metabelian, supersoluble, monomial
Aliases: C12.71D12, C6.3(D6⋊C4), (C2×C12).74D6, (C3×C12).31D4, C62.30(C2×C4), C4.Dic3.3S3, C12.27(C3⋊D4), (C6×C12).23C22, C4.13(C3⋊D12), C3⋊1(C12.47D4), C32⋊4(C4.10D4), C2.4(C6.D12), C22.3(C6.D6), (C2×C4).4S32, (C2×C6).11(C4×S3), (C2×C3⋊Dic3).1C4, (C3×C4.Dic3).2C2, (C2×C32⋊4Q8).7C2, (C3×C6).28(C22⋊C4), SmallGroup(288,209)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.71D12
G = < a,b,c | a12=1, b12=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b11 >
Subgroups: 354 in 95 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C4.10D4, C3⋊Dic3, C3×C12, C62, C4.Dic3, C3×M4(2), C2×Dic6, C3×C3⋊C8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C12.47D4, C3×C4.Dic3, C2×C32⋊4Q8, C12.71D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.10D4, S32, D6⋊C4, C6.D6, C3⋊D12, C12.47D4, C6.D12, C12.71D12
►On 48 points
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 35 45 31 41 27 37 47 33 43 29 39)(26 40 30 44 34 48 38 28 42 32 46 36)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 27 13 39)(2 32 14 44)(3 25 15 37)(4 30 16 42)(5 47 17 35)(6 28 18 40)(7 45 19 33)(8 26 20 38)(9 43 21 31)(10 48 22 36)(11 41 23 29)(12 46 24 34)
G:=sub<Sym(48)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,35,45,31,41,27,37,47,33,43,29,39)(26,40,30,44,34,48,38,28,42,32,46,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,27,13,39)(2,32,14,44)(3,25,15,37)(4,30,16,42)(5,47,17,35)(6,28,18,40)(7,45,19,33)(8,26,20,38)(9,43,21,31)(10,48,22,36)(11,41,23,29)(12,46,24,34)>;
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)(25,35,45,31,41,27,37,47,33,43,29,39)(26,40,30,44,34,48,38,28,42,32,46,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,27,13,39)(2,32,14,44)(3,25,15,37)(4,30,16,42)(5,47,17,35)(6,28,18,40)(7,45,19,33)(8,26,20,38)(9,43,21,31)(10,48,22,36)(11,41,23,29)(12,46,24,34) );
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),(25,35,45,31,41,27,37,47,33,43,29,39),(26,40,30,44,34,48,38,28,42,32,46,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,27,13,39),(2,32,14,44),(3,25,15,37),(4,30,16,42),(5,47,17,35),(6,28,18,40),(7,45,19,33),(8,26,20,38),(9,43,21,31),(10,48,22,36),(11,41,23,29),(12,46,24,34)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 36 | 36 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | C4.10D4 | S32 | C3⋊D12 | C6.D6 | C12.47D4 | C12.71D12 |
| kernel | C12.71D12 | C3×C4.Dic3 | C2×C32⋊4Q8 | C2×C3⋊Dic3 | C4.Dic3 | C3×C12 | C2×C12 | C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of C12.71D12 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 22 | 10 |
| 0 | 0 | 61 | 51 |
| 68 | 14 | 0 | 0 |
| 19 | 5 | 0 | 0 |
| 68 | 14 | 0 | 0 |
| 19 | 5 | 0 | 0 |
| 0 | 0 | 14 | 19 |
| 0 | 0 | 5 | 59 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,59,7,0,0,66,66],[0,0,68,19,0,0,14,5,22,61,0,0,10,51,0,0],[68,19,0,0,14,5,0,0,0,0,14,5,0,0,19,59] >;
C12.71D12 in GAP, Magma, Sage, TeX
C_{12}._{71}D_{12} % in TeX
G:=Group("C12.71D12"); // GroupNames label
G:=SmallGroup(288,209);
// by ID
G=gap.SmallGroup(288,209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,204,219,100,675,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^11>;
// generators/relations