Abstract: The present invention provides a camera lens consisting of eight lenses and having a small height, a wide angle, and good optical properties. The camera lens includes, sequentially from an object side, a first lens having a positive refractive power; a second lens having a positive refractive power; a third lens having a negative refractive power; a fourth lens having a positive refractive power; a fifth lens having a negative refractive power; a sixth lens having a positive refractive power; a seventh lens having a positive or negative refractive power; and an eighth lens having a negative refractive power. The camera lens satisfies specific conditions.

Abstract: An optical imaging system includes a first lens, a second lens, a third lens, a fourth lens having a negative refractive power, a fifth lens having a negative refractive power, a sixth lens, and a seventh lens having a positive refractive power. The first lens to seventh lens are sequentially disposed in a direction from an object side toward an imaging plane.

Abstract: The present disclosure discloses an optical imaging system including a first lens, a second lens, a third lens, a fourth lens, a fifth lens, a sixth lens, a seventh lens and an eighth lens, each of which has refractive power. An object-side surface of the second lens is a convex surface, and an image-side surface thereof is a concave surface; an object-side surface of the fourth lens is a convex surface, and an image-side surface thereof is a concave surface; an object-side surface of the fifth lens is a concave surface, and an image-side surface thereof is a convex surface; and each of the third lens and the sixth lens has positive refractive power. A radius of curvature R11 of an object-side surface of the sixth lens and a radius of curvature R12 of an image-side surface of the sixth lens satisfy 0.5<R11/R12<1.5.

Abstract: The present invention relates to the field of optical lenses and provides a camera optical lens. The camera optical lens includes, from an object side to an image side: a first lens; a second lens; a third lens; a fourth lens; a fifth lens; a sixth lens; a seventh lens; and an eighth lens. The camera optical lens satisfies following conditions: 4.00?f1/f?7.50; f2?0.00; and 1.55?n4?1.70, where f denotes a focal length of the camera optical lens; f1 denotes a focal length of the first lens; f2 denotes a focal length of the second lens; and n4 denotes a refractive index of the fourth lens. The present invention can achieve high optical performance while achieving ultra-thin, wide-angle lenses having a big aperture.

Abstract: Provided is a camera optical lens including, sequentially from an object side to an image side, first to seventh lenses. The camera optical lens satisfies following conditions: 20.00?R7/d7?30.00; 20.00?v7?v5?30.00; and ?12.00?(R1?R2)/(R3?R4)??1.00, where v5 denotes an abbe number of the fifth lens; v7 denotes an abbe number of the seventh lens; R1 denotes a curvature radius of an object side surface of the first lens; R2 denotes a curvature radius of an image side surface of the first lens; R3 denotes a curvature radius of an object side surface of the second lens; R4 denotes a curvature radius of an image side surface of the second lens; R7 denotes a curvature radius of an object side surface of the fourth lens; and d7 denotes an on-axis thickness of the fourth lens. The camera optical lens can achieve high optical performance such as large-aperture, wide-angle and ultra-thin.

Abstract: An imaging lens includes a first lens; a second lens; a third lens; a fourth lens; a fifth lens; a sixth lens; and a seventh lens, arranged in this order from an object side to an image plane side with a space between each of the lenses. The second lens is formed in a shape so that a surface thereof on the object side has a positive curvature radius. The third lens is formed in a shape so that a surface thereof on the image plane side has a positive curvature radius. The sixth lens is formed in a shape so that a surface thereof on the image plane side has a positive curvature radius. The seventh lens has a specific Abbe's number. The sixth lens and the seventh lens have a specific composite focal length.

Abstract: The present application discloses a camera optical lens. The camera optical lens including, in an order from an object side to an image side, a first lens, a second lens, a third lens, a fourth lens, a fifth lens, a sixth lens and a seventh lens. The camera optical lens further satisfies specific conditions.

Abstract: An imaging lens includes a first lens having positive refractive power; a second lens; a third lens; a fourth lens; a fifth lens having positive refractive power; and a sixth lens, arranged in this order from an object side to an image plane side. The second lens is formed in a shape so that a surface thereof on the object side is convex near an optical axis thereof. The fourth lens is formed in a shape so that a surface thereof on the image plane side is convex near an optical axis thereof. The fifth lens is formed in a shape so that a surface thereof on the object side is convex near an optical axis thereof. The sixth lens is formed in a shape so that a surface thereof on the image plane side is concave toward the image plane side near an optical axis thereof.

Abstract: The present disclosure discloses a camera optical lens. The camera optical lens includes, in an order from an object side to an image side, an aperture, a first lens, a second lens, a third lens, a fourth lens, and a fifth lens. The camera optical lens further satisfies specific conditions.

Abstract: An optical imaging system includes a first lens group and a second lens group. The first lens group includes a first lens and a second lens. The second lens group includes a third lens, a fourth lens, a fifth lens, a sixth lens, and a seventh lens. The first to seventh lenses are sequentially disposed from an object side toward an imaging plane. The optical imaging system satisfies TTL/2Y<1.3, where TTL is a distance from an object-side surface of the first lens to the imaging plane, and 2Y is a diagonal length of the imaging plane.

Abstract: The present disclosure discloses a camera optical lens. The camera optical lens including, in an order from an object side to an image side, a first lens, a second lens, a third lens, a fourth lens, a fifth lens, a sixth lens and a seventh lens. The first lens is made of plastic material, the second lens is made of glass material, the third lens is made of plastic material, the fourth lens is made of plastic material, the fifth lens is made of plastic material, the sixth lens is made of glass material, and the seventh lens is made of plastic material. The camera optical lens further satisfies specific conditions.

Abstract: A imaging lens assembly includes, in order from an object side to an image side, a first lens element, a second lens element, a third lens element, a fourth lens element, a fifth lens element and a sixth lens element. The first lens element has negative refractive power. The second lens element has positive refractive power. The third lens element has positive refractive power. The fourth lens element has positive refractive power. The fifth lens element has positive refractive power. The sixth lens element has negative refractive power. The imaging lens assembly has a total of six lens elements.

Abstract: An imaging lens includes a first lens having positive refractive power; a second lens having negative refractive power; a third lens having negative refractive power; a fourth lens; a fifth lens; and a sixth lens having negative refractive power, arranged in this order from an object side to an image plane side. The fourth lens and the fifth lens have a composite focal length f45, and the third lens has an Abbe's number ?d3 so that the following conditional expressions are satisfied: 0<f45, 40<?d3<75.

Abstract: Wide-angle lens comprising floating element (G2) and aperture stop (BL) comprising a group (G1) of ten lens elements with refractive power sequence arranged in a manner mirrored in sign with respect to the stop (BL), wherein lens elements having positive and negative refractive power are arranged alternately in the light direction upstream and downstream of the stop (BL) and the group (G1) is displaceable for the purpose of focusing.

Abstract: The present invention relates to an optical system. An optical system of the present invention includes, sequentially from an object side, a first lens having a positive refractive power and an object-side surface convex toward the object side; a second lens having a negative refractive power; a third lens having a negative refractive power; a fourth lens having a positive refractive power, a fifth lens having a negative refractive power and an image-side surface convex toward an image side; and a sixth lens having a negative refractive power and an image-side surface concave toward the image side.

Abstract: A photographing lens and a photographing apparatus including the photographing lens. The photographing lens includes a first lens, a second lens, a third lens, a fourth lens, and a fifth lens that are arranged sequentially from an object side. The first lens has a concave surface toward the object side and has a positive refractive power, and the second lens is formed with a meniscus shape having a concave surface toward an image side and has a negative refractive power.

Abstract: A lens for projection substantially consists of a negative first lens group, a positive second lens group and a positive third lens group in this order from a magnification side. A reduction side is telecentric, and a most magnification-side lens is an aspheric plastic lens. The third lens group consists of a negative front group and a positive rear group in this order from the magnification side with an air space therebetween. The front group consists of a negative lens and a positive lens in this order from the magnification side with an air space therebetween, and the air space being narrower than the air space between the front group and the rear group. Conditional formula (1): ?0.7<fFG3/f<0.7 about front focus fFG3 of the third lens group and focal length f of an entire system is satisfied.

Abstract: When a single lens which includes a surface having the smallest effective beam diameter or a cemented lens is designated as a reference lens, a system substantially consisting of lenses disposed closer to a magnification side than the reference lens is designated as a front group, and a system substantially consisting of lenses disposed closer to a reduction side than the reference lens is designated as a rear group, the projection lens satisfying conditional expressions (1) and (2) given below as well as conditional expressions (3) and (4): 75<2?(1), ?P<10 (2), |fM/fF|<2.0 (3), and |fM/fR|<2.0 (4).

Abstract: A macro lens and an imaging apparatus include, in the order from an object side to an image side, a first lens group, a second lens group, and a third lens group. Focusing is performed by fixing the first lens group and the third lens group, and by moving the second lens group having a positive refractive power as a whole on the optical axis in focusing from an infinite object to a close distance object. The first lens group has a plane S1 and a plane S2 of curvature radii of a same sign and satisfying a specified condition.

Abstract: An optical system WL has, in order from an object, a first lens group G1 having negative refractive power and a second lens group G2 having positive refractive power, wherein the first lens group G1 is fixed and the second lens group G2 moves upon focusing from an object at infinity to an object at a finite distance, and the second lens group G2 is formed of a front group G2a located closer to the object than an aperture stop S disposed in the second lens group G2, and a rear group G2b located closer to an image than the aperture stop S.

Abstract: What is described is a modified retrofocus-type wide-angle lens, with, as seen from the object, a front group (G1) with negative refractive power, a first rear group (Gh1) with positive refractive power displaceable along the optical axis for focusing purposes and a second rear group (Gh2) with positive refractive power facing an image plane.

Abstract: An observation optical system has a negative lens unit placed at the most object-side position; an annular prism placed on the image side of the negative lens, having a reflecting surface on the negative lens side; and an imaging lens unit arranged on the image side of the negative lens and the annular prism.

Abstract: A retrofocus wide-angle lens system is provided, wherein the entire the wide-angle lens system is divided at a position that satisfies the following condition (1) between a theoretical front lens group having a negative refractive power, and a theoretical rear lens group having a positive refractive power including the diaphragm, at a minimum focal length, and wherein a flat parallel plate is disposed at the position that satisfies the following condition (1): 1.2<|fF/f|<4.0 . . . (1), wherein fF designates the focal length of the theoretical front lens group having a negative refractive power, and f designates the focal length of the entire the wide-angle lens system.

Abstract: An objective lens includes, in order from an object side, a front group having negative refractive power, an aperture stop, and a rear group having positive refractive power, wherein the front group includes, in order from the object side, a first lens which is a negative meniscus lens with a convex surface turned to the object side and a second lens which is a negative lens with a concave surface turned to the object side; the rear group includes, in order from the object side, a positive third lens and a fourth lens made up of a positive lens and a negative lens cemented together, and the objective lens satisfies conditional expression (1) below: ?0.8<f—F/f—R<?0.3 ??(1), where f_F is a focal length of the front group, and f_R is a focal length of the rear group.

Abstract: A projection optical system including a first optical system configured to form a second image conjugate to a first image and a second optical system configured to include a reflective optical element which reflects light from the second image and to project a third image conjugate to the second image onto a projection surface is provided, wherein the first optical system has a Petzval sum with a sign opposite to a sign of a Petzval sum of the second optical system.

Abstract: A wide-angle lens system includes a negative first lens group, a negative second lens group and a positive third lens group, in this order from the object. The first lens group includes a negative meniscus lens element having the convex surface facing toward the object, and a positive lens element having a convex surface facing toward the object, in this order from the object. The first lens group includes at least one aspherical surface. The second lens group includes a negative meniscus lens element having the convex surface facing toward the object, and cemented lens elements having a positive lens element and a negative lens element, in this order from the object. The third lens group includes cemented lens elements having a negative lens element and a positive lens element, and a positive lens element having a convex surface facing toward the image, in this order from the object.

Abstract: An optical system includes a first lens unit having a negative refractive power and a second lens unit having a positive refractive power. The second lens unit includes a front lens unit, a stop, and a rear lens unit. The front lens unit includes a biconvex positive lens located closest to the image side. A back focus during focusing on an infinitely-distant object point (BF), a focal length of the entire optical system (f), a focal length of the positive lens of the front lens unit located closest to the image side (fp), a focal length of the front lens unit (f2a), a refractive power of a surface of the rear lens unit located closest to the object side (?f), and a refractive power of the entire optical system (?) satisfy the following conditions: 1.0<BF/f<3.0 0.1<fp/f2a<0.5 0.7<|?f/?|<1.5.

Abstract: Providing a retrofocus lens system and an image-taking device having small variation in aberrations upon focusing on a close object with a high imaging magnification. The retrofocus lens system includes, in order from an object, a first lens group G1 having negative refractive power, a second lens group G2 having negative refractive power, and a third lens group G3 having positive refractive power. Upon focusing from infinity to a close object, the first lens group G1, the second lens group G2, and the third lens group G3 are moved to the object such that a distance between the first lens group G1 and the second lens group G2 increases, and a distance between the second lens group G2 and the third lens group G3 decreases.

Abstract: A projection objective having a fixed focal length for digital projection includes the following arrangement of the lenses in the sequence from the image (enlargement side) to the object (reduction and illumination side): a first, negative lens, a second, negative lens, a third lens, a fourth, positive lens, a fifth lens, a sixth, positive lens, a seventh, negative lens, an eighth, positive lens, and a ninth, positive lens. By means of such an objective, the quality requirements in the field of digital projection can be met with a minimal number of lenses.

Abstract: The invention relates to a compact image-formation optical system that is diminished in a diametrical direction with as few lenses as possible while well adapting to an wide-angle arrangement having a half angle of 90° or greater, and an imaging system incorporating the same. The image-formation optical system comprises, in order from its object side, a first lens L1 that is a negative lens, a second lens L2 that is a meniscus lens concave on its object side, an apertures stop S, and a lens group G having positive refracting power.

Abstract: An ultra wide angle imaging optical system includes, in order from an object side, a first lens group, an aperture stop, and a second lens group. The first lens group includes, in order from the position closest to the object side, a negative meniscus lens element convex to the object side, a negative lens element, and a negative lens element. The second lens group includes a plurality of positive lens elements. The first lens group and the second lens group each includes at least one anamorphic surface having different optical powers with respect to a first section extending in a first direction on a flat plane substantially orthogonal to an optical axis, and with respect to a second section extending in a second direction substantially orthogonal to the first direction. The first lens group includes the lens element satisfying the following conditional formula (1): ?0.6<(Ph?Pv)*(fh+fv)<?0.

Abstract: A fisheye lens system includes a negative first lens group and a positive second lens group, which are defined with respect to a maximum air-distance between lens elements constituting the fisheye lens system. The negative first lens group includes a first negative meniscus lens element having the convex surface facing toward the object, a second negative meniscus lens element having the convex surface facing toward the object, and a negative third lens element.

Abstract: A front projection display device includes an optical engine including an illumination system, an imaging system, and projection optics. The projection optics include a first lens group of negative refractive power that has at least one aspheric surface. The projection optics output an image at a half field angle of at least 45°, where the image has substantially no distortion. For example, when the first lens group is placed at a distance of less than 1 meter from a viewing screen, the output image has a size of about 40 inches diagonal or greater, and requires substantially no keystone correction. In other aspects, the optical engine can be implemented in a wall-mounted projection system, a multimedia system, a compact integrated monitor system, and a portable projection unit.

Abstract: A wide-angle projection lens includes a first lens group of negative refractive power, the first lens group having at least one aspheric surface. The projection lens also includes a second lens group of substantially zero refractive power and a third lens group of positive refractive power. The projection lens satisfies the following three conditions: |F1/F|<4.0 (Condition (1)), |F2/F|>50 (Condition (2)) and |F3/F|<3.5 (Condition (3)). In these conditions, F is the focal length of the wide-angle projection lens; F1 is the focal length of the first lens group; F2 is the focal length of the second lens group; and F3 is the focal length of the third lens group. The wide-angle projection lens is used in a front projection display device.

Abstract: A wide angle projection lens include a first lens group of negative refractive power that has at least one aspheric surface. The wide angle projection lens can output an image at a half field angle of at least 45°, where the image has substantially no distortion and requires little or no keystone correction. The wide angle projection lens can be part of an optical engine that can be implemented in rear projection or front projection display devices.

Abstract: The present invention provides a bright image reading lens coping with high resolution and yet having a great depth of focus and a wide angle of field, and an image reading apparatus using the same. The image reading lens includes, in succession from the object side, six lenses, i.e., a meniscus-shaped positive first lens having its convex surface facing the object side, a second lens of which the both lens surfaces are concave surfaces, a third lens of which the both lens surfaces are convex surfaces, a meniscus-shaped fourth lens having its concave surface facing the object side, a meniscus-shaped negative fifth lens having its concave surface facing the object side, and a meniscus-shaped negative sixth lens having its concave surface facing the object side, and satisfies a predetermined condition.

Abstract: Telecentric projection lenses for use with pixelized panels (PP) are provided. The projection lenses have a negative first unit (U1) which has at least one negative meniscus element (N1) having at least one aspheric surface and a positive second unit (U2) which has at least one positive element (P1) having at least one aspheric surface. The lens' aperture stop (AS) is located between the two units, and a third lens unit (U3) may be used in the vicinity of the aperture stop to improve the correction of axial color. The lenses have small forward vertex distances, small clear apertures, and long back focal lengths which make them particularly well-suited for use in the manufacture of compact projection systems.

Abstract: A wide-angle lens system includes a negative first lens group, a positive second lens group, a diaphragm, and a positive third lens group. Upon focusing, the positive second lens group moves along the optical axis direction.

Abstract: A lens system has the first and second lens components from the object side. The first lens component has a negative optical power. The second lens component has a positive optical power. The lens system fulfills the conditions: 0.0<PRL/Y′<0.5, 0.5<Pf/f<1.0. In the conditions, PRL represents a distance in a direction perpendicular to the optical axis between the optical axis and an incident position where a lower ray of a most off-axial rays enters said second lens component, Y′ represents a largest image height, f represents a focal length of the entire lens system, and Pf represents a focal length of said second lens component.

Abstract: A retrofocus lens system comprises a first lens group having a negative power, a second lens group having a positive power, and a third lens group having a positive power. The first lens group includes a first lens, a meniscus-shaped second lens having a negative power and a convex surface facing the large conjugate side, a third lens having a negative power, and a fourth lens. The second lens group includes a fifth lens having a positive power. The third lens group includes a sixth lens having a negative power, both surfaces of which are concave, a seventh lens having a positive power, both surfaces of which are convex, an eighth lens having a positive power, both surfaces of which are convex, and a ninth lens having a positive axial power. The system satisfies 0.9<f2/f3<1.8, 1.5<|f1|/f<2.

Abstract: A projection optical system has, from the enlargement side to the reduction side, a first, a second, and a third lens unit, and an aperture diaphragm. The negatively-powered first lens unit includes only a negative lens element and has at least one aspherical surface. The positively-powered second lens unit includes at least one positive lens element. The positively-powered third lens unit includes a lens element. The aperture diaphragm is disposed at the front focal point of the third lens unit. The lens element of the third lens unit is made of a material of which anomalous partial dispersibility and Abbe number are defined.

Abstract: In an imaging lens unit configured of two lenses, the deterioration of the optical performance due to the eccentricity is suppressed for an improved assembling property, while at the same time correcting the various aberrations in satisfactory manner. A meniscus first lens (L1) having a weak refracting power, and a meniscus second lens (L2) having positive refracting power with the convex surface thereof directed toward the image surface are arranged sequentially from the object side toward the image surface (IMG). The conditions f2/f>0.9 and nd2>1.55 are met, where f is the focal length of the whole lens system, f2 the focal length of the second lens, and nd2 the refractive index on the d line of the second lens.

Abstract: An optical system focuses an object upon an image surface of a solid-state image device. The optical system has a master optical system and a condenser lens. The master optical system and the condenser lens both have positive optical power. The focusing optical system fulfills the following conditions: ##EQU1## where a represents the distance between the exit pupil of the master optical system and the image surface;b represents the distance between the exit pupil of the entire optical system and the image surface; andY' represents the maximum image height.

Abstract: An imaging lens device at f/4 has zoomlike variable chromatic and spherochromatic correction for imagery in each of the astronomical U, B, V, and R bands. This lens system can thus take advantage of the full usable wavelength range of CCD cameras extending from 0.35-0.9 micrometers. A first triplet lens group effects the chromatic correction by altering separation between lens elements of the triplet while a second triplet lens group maintains a sharp image focus.

Abstract: A zoom lens system for use in a camera having a color filter and a CCD achieves a magnification ratio of about 2.9 or greater and a back focus of about 9.2 or greater. The zoom lens system includes in order from an object side of the lens system, a first lens group having a negative refracting power, a second lens group spaced from the first lens group and having a positive refracting power and a third lens group spaced from the second lens group and having a positive refracting power. One or both of the fifth and an eighth lens surface has an aspherical surface, the fifth and eighth lens surfaces being counted from the object side. The second lens group includes five lenses with an extra lens being located at a far end of the second lens group relative to the object side. Each of the first, second and third lens groups is movable from a first position when the lens system is in a wide position to a second position when the lens system is in a tele position.

Abstract: A wide-angle lens with an image stabilizing function comprises, from a front, object side to a rear, image side, a lens group GF with negative refractive power; a lens group GL with positive refractive power, wherein the back focal length is longer than the focal length of the entire optical system and overall refractive power is positive and wherein a partial lens group GLP having positive refractive power out of the lens group GL comprises at least a positive meniscus lens with the convex surface facing the image side and a biconvex lens; and a displacement device provided to reduce vibration by causing rotative motion of the partial lens group GLP about a point on the optical axis on the object side which is separated by a predetermined distance from a surface of the partial lens group GLP closest to the image side, wherein the focal length fLP of the partial lens group GLP and the focal length f of the whole lens system at infinite objective distance satisfy 0.3<fLP/f<5.0.

Abstract: A wide angle lens is made of three groups so as to satisfy a predetermined conditional expression concerning abbe number, thereby making it telecentric and yielding no eclipse in its peripheral luminous flux. Also, an apparatus accommodating it is made compact as a whole. This wide angle lens comprises, arranged in the following order from the screen side toward the liquid crystal side, a first lens group composed of a concave lens (L1), whose concave surface having a large radius of curvature is directed toward a smaller conjugate side (liquid crystal panel), and a convex lens (L2); a second lens group composed of one concave lens (L3) and two convex lenses (L4) and (L5); and a third lens group composed of a Fresnel lens (L6) which is disposed near a liquid crystal panel (L7; 4). The abbe number .nu. 1 of the lens L1 which is the concave lens in the first lens group is 57.8, whereas the abbe number .nu. 2 of the lens (L3) which is the concave lens in the second lens group is 25.

Abstract: A projection system wherein two or more separate display sources, such as high brightness sources, or active matrix liquid crystal display (AMLCD) or other flat panel displays are used to write information directly onto a display screen, or onto a photoactivated reflective light valve for projection onto the display screen. Sequential imaging apparatus is used to alternately generate first and second images for display. For example, such images are coupled to the photoactivated reflective light valve and an illumination source is employed to illuminate the light valve and project a composite image onto the display screen. The flat panel displays may be transmissive, reflective, or self-emissive with the use of an optical shutter. A writing illumination source is used for non-self-emissive flat panel displays. Using high brightness display sources, the images may be directly written onto the display screen without the use of the reflective photoactivated light valve and the illumination source.

Abstract: A wide-angle fast lens system includes a front lens group having a negative power, a diaphragm and a rear lens group having a positive power. The front lens group consists of a first sub-lens group and a second sub-lens group having a negative power. The lens system satisfies the following relationships:-0.6<f/f.sub.F <-0.2,6.0<.SIGMA.d.sub.F+S /f<12.0,f/.vertline.f.sub.1a .vertline.<0.07, and0.4<h.sub.aF /h.sub.aL <0.65,wherein "f" represents the focal length of the whole lens system, "f.sub.F " represents the focal length of the front lens group, ".SIGMA.d.sub.F+S " represents the sum of the thickness of the front lens group and the distance between the front and rear lens groups, "f.sub.1a " represents the focal length of the first sub-lens group, "h.sub.aF " represents the height of the paraxial rays incident on the first surface of the first sub-lens group, and "h.sub.aL " represents the height of the paraxial rays incident on the last surface of the first sub-lens group.

Abstract: A retrofocus type standard lens includes a front lens group having a positive power, a diaphragm, and a rear lens group having a negative power, in this order from the object side. The focusing is carried out by moving the front lens group without moving the diaphragm and the rear lens group. The front lens group has a negative subgroup and a positive subgroup. The retrofocus type standard lens satisfies the relationships:0.5<f/f.sub.F <1.0;-0.7<f/f.sub.FN <-0.3,wherein "f" designates a focal length of an entire lens system; "f" designates a focal length of the front lens group; and "f.sub.FN " designates a focal length of the negative subgroup.