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G = C12.70D12  order 288 = 25·32

1st non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.70D12, C6.2(D6:C4), (C2xC12).72D6, (C3xC12).29D4, C4.Dic3:3S3, C62.28(C2xC4), C12.25(C3:D4), (C6xC12).21C22, C32:3(C4.D4), C4.12(C3:D12), C3:1(C12.46D4), C2.3(C6.D12), C22.2(C6.D6), (C2xC4).2S32, (C2xC6).10(C4xS3), (C22xC3:S3).1C4, (C2xC12:S3).7C2, (C3xC4.Dic3):14C2, (C3xC6).26(C22:C4), SmallGroup(288,207)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12.70D12
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C6xC12 — C3xC4.Dic3 — C12.70D12
Lower central
C32 — C3xC6 — C62 — C12.70D12
Upper central
C1 — C2 — C2xC4

Generators and relations for C12.70D12
 G = < a,b,c | a12=c2=1, b12=a6, bab-1=cac=a-1, cbc=a9b11 >

Subgroups: 706 in 119 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2xC4, D4, C23, C32, C12, C12, D6, C2xC6, C2xC6, M4(2), C2xD4, C3:S3, C3xC6, C3xC6, C3:C8, C24, D12, C2xC12, C2xC12, C22xS3, C4.D4, C3xC12, C2xC3:S3, C62, C4.Dic3, C3xM4(2), C2xD12, C3xC3:C8, C12:S3, C6xC12, C22xC3:S3, C12.46D4, C3xC4.Dic3, C2xC12:S3, C12.70D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, C4.D4, S32, D6:C4, C6.D6, C3:D12, C12.46D4, C6.D12, C12.70D12

Permutation representations of C12.70D12
►On 24 points - transitive group 24T617
Generators in S24
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)

(1 15)(2 20)(3 13)(4 18)(5 11)(6 16)(7 9)(8 14)(10 12)(17 23)(19 21)(22 24)
 
G:=sub<Sym(24)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,11)(6,16)(7,9)(8,14)(10,12)(17,23)(19,21)(22,24)>;
 
G:=Group( (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,11)(6,16)(7,9)(8,14)(10,12)(17,23)(19,21)(22,24) );
 
G=PermutationGroup([[(1,11,21,7,17,3,13,23,9,19,5,15),(2,16,6,20,10,24,14,4,18,8,22,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15),(2,20),(3,13),(4,18),(5,11),(6,16),(7,9),(8,14),(10,12),(17,23),(19,21),(22,24)]])
 
G:=TransitiveGroup(24,617);
 

39 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B6A6B6C···6G8A8B8C8D12A12B12C12D12E···12J24A···24H
order1222233344666···688881212121212···1224···24
size112363622422224···41212121222224···412···12

39 irreducible representations

dim1111222222444444
type+++++++++++++
imageC1C2C2C4S3D4D6D12C3:D4C4xS3C4.D4S32C3:D12C6.D6C12.46D4C12.70D12
kernelC12.70D12C3xC4.Dic3C2xC12:S3C22xC3:S3C4.Dic3C3xC12C2xC12C12C12C2xC6C32C2xC4C4C22C3C1
# reps1214222444112144

Matrix representation of C12.70D12 ►in GL4(F73) generated by

14700
66700
00147
00667
,
005966
00714
727200
0100
,
727200
0100
00072
00720
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,72,0,0,0,72,1,59,7,0,0,66,14,0,0],[72,0,0,0,72,1,0,0,0,0,0,72,0,0,72,0] >;
 

C12.70D12 in GAP, Magma, Sage, TeX 

C_{12}._{70}D_{12}
 
% in TeX
 
G:=Group("C12.70D12");
 
// GroupNames label
 
G:=SmallGroup(288,207);
 
// by ID
 
G=gap.SmallGroup(288,207);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,100,675,346,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=c^2=1,b^12=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^9*b^11>;
 
// generators/relations
 

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