Linear and projective linear groups

Over any field K, linear group usually refers to an algebraic group which is a (Zariski closed) subgroup of the general linear group GL(n,K) of invertible nxn matrices with coefficients in K. Similarly, projective linear group refers to a closed subgroup of PGL(n,K)=GL(n,K)/{scalar matrices}, the group of linear automorphisms of the projective n-space. When K=Fq is a finite field, these groups are finite, and often simple or almost simple, and so they play an important role in the classification of finite simple groups.

Groups of order 6

LabeldρID
S3GL2(F2)General linear group on F22S332+6,1
SL2(F2)Special linear group on F22
PGL2(F2)Projective linear group on F22
PSL2(F2)Projective special linear group on F22
SO3(F2)Special orthogonal group on F23
SU2(F2)Special unitary group on F22
O3(F2)Orthogonal group on F23
PSO3(F2)Projective special orthogonal group on F23
PO3(F2)Projective orthogonal group on F23
AGL1(F3)Affine linear group on F31
PU2(F2)Projective unitary group on F22
PSU2(F2)Projective special unitary group on F22
CO3(F2)Conformal orthogonal group on F23
CSU2(F2)Conformal special unitary group on F22
CSO3(F2)Conformal special orthogonal group on F23
Spin3(F2)Spin group on F23
Ω3(F2)Omega group on F23
3(F2)Projective Omega group on F23

Groups of order 8

LabeldρID
D4AΣL1(F4)Affine special semilinear group on F41D442+8,3

Groups of order 12

LabeldρID
A4PSL2(F3)Projective special linear group on F32A443+12,3
AGL1(F4)Affine linear group on F41
PSU2(F3)Projective special unitary group on F32
Ω3(F3)Omega group on F33
3(F3)Projective Omega group on F33

Groups of order 18

LabeldρID
C3xS3U2(F2)Unitary group on F22C3xS36218,3
AΣL1(F9)Affine special semilinear group on F91
CU2(F2)Conformal unitary group on F22

Groups of order 20

LabeldρID
F5AGL1(F5)Affine linear group on F51F554+20,3

Groups of order 24

LabeldρID
SL2(F3)SL2(F3)Special linear group on F32SL(2,3)82-24,3
SU2(F3)Special unitary group on F32
Spin3(F3)Spin group on F33
S4PGL2(F3)Projective linear group on F32S443+24,12
SO3(F3)Special orthogonal group on F33
PSO3(F3)Projective special orthogonal group on F33
PO3(F3)Projective orthogonal group on F33
AGL2(F2)Affine linear group on F22
ASL2(F2)Affine special linear group on F22
AΓL1(F4)Affine semilinear group on F41
PU2(F3)Projective unitary group on F32
CSO3(F3)Conformal special orthogonal group on F33
C2xA4AΣL1(F8)Affine special semilinear group on F81C2xA463+24,13

Groups of order 36

LabeldρID
S32Spin4+ (F2)Spin group of + type on F24S3^264+36,10
Ω4+ (F2)Omega group of + type on F24
4+ (F2)Projective Omega group of + type on F24

Groups of order 42

LabeldρID
F7AGL1(F7)Affine linear group on F71F776+42,1

Groups of order 48

LabeldρID
CSU2(F3)CSU2(F3)Conformal special unitary group on F32CSU(2,3)162-48,28
GL2(F3)GL2(F3)General linear group on F32GL(2,3)8248,29
C2xS4O3(F3)Orthogonal group on F33C2xS463+48,48
CO3(F3)Conformal orthogonal group on F33

Groups of order 50

LabeldρID
C5xD5AΣL1(F25)Affine special semilinear group on F251C5xD510250,3

Groups of order 56

LabeldρID
F8AGL1(F8)Affine linear group on F81F887+56,11

Groups of order 60

LabeldρID
A5SL2(F4)Special linear group on F42A553+60,5
PGL2(F4)Projective linear group on F42
PSL2(F5)Projective special linear group on F52
PSL2(F4)Projective special linear group on F42
SO3(F4)Special orthogonal group on F43
SU2(F4)Special unitary group on F42
O3(F4)Orthogonal group on F43
PSO3(F4)Projective special orthogonal group on F43
PO3(F4)Projective orthogonal group on F43
PU2(F4)Projective unitary group on F42
PSU2(F5)Projective special unitary group on F52
PSU2(F4)Projective special unitary group on F42
CSU2(F4)Conformal special unitary group on F42
Spin3(F4)Spin group on F43
Ω3(F5)Omega group on F53
Ω3(F4)Omega group on F43
Ω4- (F2)Omega group of - type on F24
3(F4)Projective Omega group on F43
3(F5)Projective Omega group on F53
4- (F2)Projective Omega group of - type on F24

Groups of order 64

LabeldρID
C2wrC4AΣL1(F16)Affine special semilinear group on F161C2wrC484+64,32

Groups of order 72

LabeldρID
F9AGL1(F9)Affine linear group on F91F998+72,39
S3wrC2SO4+ (F2)Special orthogonal group of + type on F24S3wrC264+72,40
O4+ (F2)Orthogonal group of + type on F24
PSO4+ (F2)Projective special orthogonal group of + type on F24
PO4+ (F2)Projective orthogonal group of + type on F24
CO4+ (F2)Conformal orthogonal group of + type on F24
CSO4+ (F2)Conformal special orthogonal group of + type on F24
PCO4+ (F2)Projective conformal orthogonal group of + type on F24
PCSO4+ (F2)Projective conformal special orthogonal group of + type on F24
PSU3(F2)PSU3(F2)Projective special unitary group on F23PSU(3,2)98+72,41

Groups of order 81

LabeldρID
C3wrC3AΣL1(F27)Affine special semilinear group on F271C3wrC39381,7

Groups of order 96

LabeldρID
U2(F3)U2(F3)Unitary group on F32U(2,3)24296,67

Groups of order 98

LabeldρID
C7xD7AΣL1(F49)Affine special semilinear group on F491C7xD714298,3

Groups of order 110

LabeldρID
F11AGL1(F11)Affine linear group on F111F111110+110,1

Groups of order 120

LabeldρID
SL2(F5)SL2(F5)Special linear group on F52SL(2,5)242-120,5
SU2(F5)Special unitary group on F52
Spin3(F5)Spin group on F53
S5PGL2(F5)Projective linear group on F52S554+120,34
SO3(F5)Special orthogonal group on F53
SO4- (F2)Special orthogonal group of - type on F24
O4- (F2)Orthogonal group of - type on F24
PSO3(F5)Projective special orthogonal group on F53
PSO4- (F2)Projective special orthogonal group of - type on F24
PO3(F5)Projective orthogonal group on F53
PO4- (F2)Projective orthogonal group of - type on F24
PU2(F5)Projective unitary group on F52
PΓL2(F4)Projective semilinear group on F42
PΣL2(F4)Projective special semilinear group on F42
PΓU2(F4)Projective semilinear unitary group on F42
PΣU2(F4)Projective semilinear special unitary group on F42
CO4- (F2)Conformal orthogonal group of - type on F24
CSO3(F5)Conformal special orthogonal group on F53
CSO4- (F2)Conformal special orthogonal group of - type on F24
PCO4- (F2)Projective conformal orthogonal group of - type on F24
PCSO4- (F2)Projective conformal special orthogonal group of - type on F24
ΣL2(F4)Special semilinear group on F42

Groups of order 144

LabeldρID
AΓL1(F9)AΓL1(F9)Affine semilinear group on F91AGammaL(1,9)98+144,182
A424+ (F3)Projective Omega group of + type on F34A4^2129+144,184

Groups of order 156

LabeldρID
F13AGL1(F13)Affine linear group on F131F131312+156,7

Groups of order 160

LabeldρID
C2xC24:C5AΣL1(F32)Affine special semilinear group on F321C2xC2^4:C5105+160,235

Groups of order 168

LabeldρID
GL3(F2)GL3(F2)General linear group on F23GL(3,2)73168,42
SL3(F2)Special linear group on F23
PGL3(F2)Projective linear group on F23
PSL3(F2)Projective special linear group on F23
PSL2(F7)Projective special linear group on F72
PSU2(F7)Projective special unitary group on F72
Ω3(F7)Omega group on F73
3(F7)Projective Omega group on F73
AΓL1(F8)AΓL1(F8)Affine semilinear group on F81AGammaL(1,8)87+168,43

Groups of order 180

LabeldρID
C3xA5GL2(F4)General linear group on F42C3xA5153180,19
CO3(F4)Conformal orthogonal group on F43
CSO3(F4)Conformal special orthogonal group on F43

Groups of order 192

LabeldρID
CU2(F3)CU2(F3)Conformal unitary group on F32CU(2,3)322192,963

Groups of order 216

LabeldρID
SU3(F2)SU3(F2)Special unitary group on F23SU(3,2)273216,88
CSU3(F2)Conformal special unitary group on F23
ASL2(F3)ASL2(F3)Affine special linear group on F32ASL(2,3)98+216,153
PU3(F2)Projective unitary group on F23

Groups of order 240

LabeldρID
CSU2(F5)CSU2(F5)Conformal special unitary group on F52CSU(2,5)484-240,89
C2xS5O3(F5)Orthogonal group on F53C2xS5104+240,189
F16AGL1(F16)Affine linear group on F161F161615+240,191

Groups of order 242

LabeldρID
C11xD11AΣL1(F121)Affine special semilinear group on F1211C11xD11222242,3

Groups of order 272

LabeldρID
F17AGL1(F17)Affine linear group on F171F171716+272,50

Groups of order 288

LabeldρID
Ω4+ (F3)Ω4+ (F3)Omega group of + type on F34Omega+(4,3)244+288,860
PSO4+ (F3)PSO4+ (F3)Projective special orthogonal group of + type on F34PSO+(4,3)129+288,1026

Groups of order 300

LabeldρID
C5xA5U2(F4)Unitary group on F42C5xA5253300,22

Groups of order 324

LabeldρID
C3xC33:C4AΣL1(F81)Affine special semilinear group on F811C3xC3^3:C4124324,162

Groups of order 336

LabeldρID
SL2(F7)SL2(F7)Special linear group on F72SL(2,7)164336,114
SU2(F7)Special unitary group on F72
Spin3(F7)Spin group on F73
PGL2(F7)PGL2(F7)Projective linear group on F72PGL(2,7)86+336,208
SO3(F7)Special orthogonal group on F73
PSO3(F7)Projective special orthogonal group on F73
PO3(F7)Projective orthogonal group on F73
PU2(F7)Projective unitary group on F72

Groups of order 338

LabeldρID
C13xD13AΣL1(F169)Affine special semilinear group on F1691C13xD13262338,3

Groups of order 342

LabeldρID
F19AGL1(F19)Affine linear group on F191F191918+342,7

Groups of order 360

LabeldρID
A6PSL2(F9)Projective special linear group on F92A665+360,118
PSO4- (F3)Projective special orthogonal group of - type on F34
PSU2(F9)Projective special unitary group on F92
Ω3(F9)Omega group on F93
Ω4- (F3)Omega group of - type on F34
3(F9)Projective Omega group on F93
4- (F3)Projective Omega group of - type on F34
ΓL2(F4)ΓL2(F4)Semilinear group on F42GammaL(2,4)156360,120

Groups of order 375

LabeldρID
C5xC52:C3AΣL1(F125)Affine special semilinear group on F1251C5xC5^2:C3153375,6

Groups of order 432

LabeldρID
AGL2(F3)AGL2(F3)Affine linear group on F32AGL(2,3)98+432,734

Groups of order 480

LabeldρID
GL2(F5)GL2(F5)General linear group on F52GL(2,5)244480,218
C4xS5CO3(F5)Conformal orthogonal group on F53C4xS5204480,943
׿
x
:
Z
F
o
wr
Q
<