metabelian, supersoluble, monomial, A-group
Aliases: C12.29D6, C3⋊C8⋊6S3, C3⋊S3⋊2C8, C3⋊1(S3×C8), C4.14S32, C6.1(C4×S3), C32⋊4(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
| C32 — C12.29D6 |
Generators and relations for C12.29D6
G = < a,b,c | a12=1, b6=a3, c2=a6, bab-1=cac-1=a5, cbc-1=a6b5 >
Permutation representations of C12.29D6
►On 24 points - transitive group 24T237
(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⋊C8⋊20D6 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 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | C4×S3 | S3×C8 | S32 | C6.D6 | C12.29D6 |
| kernel | C12.29D6 | C3×C3⋊C8 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C12.29D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 1 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 2 | 0 | 1 |
| 4 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
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
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