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G = C12.70D12order 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), (C2×C12).72D6, (C3×C12).29D4, C4.Dic33S3, C62.28(C2×C4), C12.25(C3⋊D4), (C6×C12).21C22, C323(C4.D4), C4.12(C3⋊D12), C31(C12.46D4), C2.3(C6.D12), C22.2(C6.D6), (C2×C4).2S32, (C2×C6).10(C4×S3), (C22×C3⋊S3).1C4, (C2×C12⋊S3).7C2, (C3×C4.Dic3)⋊14C2, (C3×C6).26(C22⋊C4), SmallGroup(288,207)

Series: Derived Chief Lower central Upper central

C1C62 — C12.70D12
C1C3C32C3×C6C3×C12C6×C12C3×C4.Dic3 — C12.70D12
C32C3×C6C62 — C12.70D12
C1C2C2×C4

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 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×8], C6 [×2], C6 [×5], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×16], C2×C6 [×2], C2×C6, M4(2) [×2], C2×D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], D12 [×8], C2×C12 [×2], C2×C12, C22×S3 [×6], C4.D4, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C4.Dic3 [×2], C3×M4(2) [×2], C2×D12 [×3], C3×C3⋊C8 [×2], C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C12.46D4 [×2], C3×C4.Dic3 [×2], C2×C12⋊S3, C12.70D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C4.D4, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C12.46D4 [×2], 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⋊D4C4×S3C4.D4S32C3⋊D12C6.D6C12.46D4C12.70D12
kernelC12.70D12C3×C4.Dic3C2×C12⋊S3C22×C3⋊S3C4.Dic3C3×C12C2×C12C12C12C2×C6C32C2×C4C4C22C3C1
# 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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