# 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 n×n 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
C3×S3U2(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
C2×A4AΣL1(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
C2×S4O3(𝔽3)Orthogonal group on 𝔽33C2xS463+48,48
CO3(𝔽3)Conformal orthogonal group on 𝔽33

### Groups of order 50

LabeldρID
C5×D5AΣL1(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
C2≀C4AΣL1(𝔽16)Affine special semilinear group on 𝔽161C2wrC484+64,32

### Groups of order 72

LabeldρID
F9AGL1(𝔽9)Affine linear group on 𝔽91F998+72,39
S3≀C2SO4+ (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
C3≀C3AΣL1(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
C7×D7AΣL1(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
C2×C24⋊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
C3×A5GL2(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
C2×S5O3(𝔽5)Orthogonal group on 𝔽53C2xS5104+240,189
F16AGL1(𝔽16)Affine linear group on 𝔽161F161615+240,191

### Groups of order 242

LabeldρID
C11×D11AΣL1(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
C5×A5U2(𝔽4)Unitary group on 𝔽42C5xA5253300,22

### Groups of order 324

LabeldρID
C3×C33⋊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
C13×D13AΣL1(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
C5×C52⋊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
C4×S5CO3(𝔽5)Conformal orthogonal group on 𝔽53C4xS5204480,943
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