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G = C12.29D6order 144 = 24·32

3rd non-split extension by C12 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C12.29D6, C3⋊C86S3, C3⋊S32C8, C31(S3×C8), C4.14S32, C6.1(C4×S3), C324(C2×C8), C3⋊Dic3.2C4, (C3×C12).28C22, C2.1(C6.D6), (C3×C3⋊C8)⋊6C2, (C2×C3⋊S3).2C4, (C4×C3⋊S3).3C2, (C3×C6).9(C2×C4), SmallGroup(144,53)

Series: Derived Chief Lower central Upper central

C1C32 — C12.29D6
C1C3C32C3×C6C3×C12C3×C3⋊C8 — C12.29D6
C32 — C12.29D6
C1C4

Generators and relations for C12.29D6
 G = < a,b,c | a12=1, b6=a3, c2=a6, bab-1=cac-1=a5, cbc-1=a6b5 >

9C2
9C2
2C3
9C22
9C4
2C6
3S3
3S3
3S3
3S3
6S3
6S3
3C8
3C8
9C2×C4
2C12
3D6
3Dic3
3D6
3Dic3
6D6
6Dic3
9C2×C8
3C24
3C24
3C4×S3
3C4×S3
6C4×S3
3S3×C8
3S3×C8

Permutation representations of C12.29D6
On 24 points - transitive group 24T237
Generators in S24
(1 11 21 7 17 3 13 23 9 19 5 15)(2 4 6 8 10 12 14 16 18 20 22 24)
(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 13 3)(2 8 14 20)(4 18 16 6)(5 11 17 23)(7 21 19 9)(10 24 22 12)

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

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

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

G:=TransitiveGroup(24,237);

C12.29D6 is a maximal subgroup of
S32⋊C8  C3⋊S3.2D8  C3⋊S3.2Q16  C32⋊C4⋊C8  S32×C8  C24⋊D6  C24.64D6  C3⋊C8.22D6  C3⋊C820D6  D12⋊D6  Dic6⋊D6  D12.8D6  D12.9D6  Dic6.9D6  D12.14D6  C36.38D6  C32⋊C6⋊C8  C12.69S32  C12.93S32
C12.29D6 is a maximal quotient of
C24.60D6  C24.62D6  C6.(S3×C8)  C12.78D12  C12.15Dic6  C36.38D6  C12.89S32  C12.69S32  C12.93S32

36 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C8A···8H12A12B12C12D12E12F24A···24H
order122233344446668···812121212121224···24
size119922411992243···32222446···6

36 irreducible representations

dim1111112222444
type+++++++
imageC1C2C2C4C4C8S3D6C4×S3S3×C8S32C6.D6C12.29D6
kernelC12.29D6C3×C3⋊C8C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C3⋊S3C3⋊C8C12C6C3C4C2C1
# reps1212282248112

Matrix representation of C12.29D6 in GL4(𝔽5) generated by

0010
0201
1020
0100
,
0201
4030
0400
2000
,
0010
0004
4000
0100
G:=sub<GL(4,GF(5))| [0,0,1,0,0,2,0,1,1,0,2,0,0,1,0,0],[0,4,0,2,2,0,4,0,0,3,0,0,1,0,0,0],[0,0,4,0,0,0,0,1,1,0,0,0,0,4,0,0] >;

C12.29D6 in GAP, Magma, Sage, TeX

C_{12}._{29}D_6
% in TeX

G:=Group("C12.29D6");
// GroupNames label

G:=SmallGroup(144,53);
// by ID

G=gap.SmallGroup(144,53);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,31,50,490,3461]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^6=a^3,c^2=a^6,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^5>;
// generators/relations

Export

Subgroup lattice of C12.29D6 in TeX

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