Non-soluble groups

See soluble groups.

Groups of order 60

dρLabelID
A5Alternating group on 5 letters; = SL2(F4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple53+A560,5

Groups of order 120

dρLabelID
S5Symmetric group on 5 letters; = PGL2(F5) = Aut(A5) = 5-cell symmetries; almost simple54+S5120,34
SL2(F5)Special linear group on F52; = C2.A5 = 2I = <2,3,5>242-SL(2,5)120,5
C2xA5Direct product of C2 and A5; = icosahedron/dodecahedron symmetries103+C2xA5120,35

Groups of order 168

dρLabelID
GL3(F2)General linear group on F23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simple73GL(3,2)168,42

Groups of order 180

dρLabelID
C3xA5Direct product of C3 and A5; = GL2(F4)153C3xA5180,19

Groups of order 240

dρLabelID
CSU2(F5)Conformal special unitary group on F52; = C2.2S5484-CSU(2,5)240,89
A5:C4The semidirect product of A5 and C4 acting via C4/C2=C2124A5:C4240,91
C4.A5The central extension by C4 of A5242C4.A5240,93
C2.S52nd central stem extension by C2 of S5404-C2.S5240,90
C2xS5Direct product of C2 and S5; = O3(F5)104+C2xS5240,189
C4xA5Direct product of C4 and A5203C4xA5240,92
C22xA5Direct product of C22 and A520C2^2xA5240,190
C2xSL2(F5)Direct product of C2 and SL2(F5)48C2xSL(2,5)240,94

Groups of order 300

dρLabelID
C5xA5Direct product of C5 and A5; = U2(F4)253C5xA5300,22

Groups of order 336

dρLabelID
SL2(F7)Special linear group on F72; = C2.GL3(F2)164SL(2,7)336,114
PGL2(F7)Projective linear group on F72; = GL3(F2):C2 = Aut(GL3(F2)); almost simple86+PGL(2,7)336,208
C2xGL3(F2)Direct product of C2 and GL3(F2)143C2xGL(3,2)336,209

Groups of order 360

dρLabelID
A6Alternating group on 6 letters; = PSL2(F9) = L2(9); 3rd non-abelian simple65+A6360,118
ΓL2(F4)Semilinear group on F42; = C3:S5156GammaL(2,4)360,120
C3xS5Direct product of C3 and S5154C3xS5360,119
S3xA5Direct product of S3 and A5156+S3xA5360,121
C6xA5Direct product of C6 and A5303C6xA5360,122
C3xSL2(F5)Direct product of C3 and SL2(F5)722C3xSL(2,5)360,51

Groups of order 420

dρLabelID
C7xA5Direct product of C7 and A5353C7xA5420,13

Groups of order 480

dρLabelID
GL2(F5)General linear group on F52; = SL2(F5):1C4 = Aut(C52)244GL(2,5)480,218
C4:S5The semidirect product of C4 and S5 acting via S5/A5=C2206C4:S5480,944
C22:S5The semidirect product of C22 and S5 acting via S5/A5=C2206+C2^2:S5480,951
A5:Q8The semidirect product of A5 and Q8 acting via Q8/C4=C2246A5:Q8480,945
A5:C8The semidirect product of A5 and C8 acting via C8/C4=C2404A5:C8480,217
C4.3S53rd non-split extension by C4 of S5 acting via S5/A5=C2404C4.3S5480,948
C8.A5The central extension by C8 of A5482C8.A5480,221
D4.A5The non-split extension by D4 of A5 acting through Inn(D4)484-D4.A5480,957
Q8.A5The non-split extension by Q8 of A5 acting through Inn(Q8)484+Q8.A5480,959
C4.6S53rd central extension by C4 of S5484C4.6S5480,946
C4.S52nd non-split extension by C4 of S5 acting via S5/A5=C2484C4.S5480,947
C22.S5The non-split extension by C22 of S5 acting via S5/A5=C2484-C2^2.S5480,953
C22.2S51st central extension by C22 of S596C2^2.2S5480,219
C4xS5Direct product of C4 and S5; = CO3(F5)204C4xS5480,943
D4xA5Direct product of D4 and A5206+D4xA5480,956
C22xS5Direct product of C22 and S520C2^2xS5480,1186
C8xA5Direct product of C8 and A5403C8xA5480,220
Q8xA5Direct product of Q8 and A5406-Q8xA5480,958
C23xA5Direct product of C23 and A540C2^3xA5480,1187
C4xSL2(F5)Direct product of C4 and SL2(F5)96C4xSL(2,5)480,222
C2xCSU2(F5)Direct product of C2 and CSU2(F5)96C2xCSU(2,5)480,949
C22xSL2(F5)Direct product of C22 and SL2(F5)96C2^2xSL(2,5)480,960
C2xA5:C4Direct product of C2 and A5:C424C2xA5:C4480,952
C2xC4xA5Direct product of C2xC4 and A540C2xC4xA5480,954
C2xC4.A5Direct product of C2 and C4.A548C2xC4.A5480,955
C2xC2.S5Direct product of C2 and C2.S580C2xC2.S5480,950
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