metabelian, supersoluble, monomial
Aliases: C12.8Dic6, C62.29D4, (C3×C6).17D8, (C3×C6).7Q16, (C3×C12).8Q8, C12.22(C4×S3), C32⋊4C8⋊2C4, (C2×C12).83D6, C6.12(D4⋊S3), C32⋊5(C2.D8), C3⋊2(C6.Q16), C4⋊Dic3.10S3, C6.5(C3⋊Q16), C6.6(Dic3⋊C4), (C6×C12).36C22, C4.5(C32⋊2Q8), C4.7(C6.D6), C2.2(C32⋊2D8), C2.2(C32⋊2Q16), C2.4(C62.C22), C22.10(D6⋊S3), (C2×C4).103S32, (C3×C6).23(C4⋊C4), (C3×C12).34(C2×C4), (C3×C4⋊Dic3).6C2, (C2×C6).52(C3⋊D4), (C2×C32⋊4C8).3C2, SmallGroup(288,224)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.8Dic6
G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, cac-1=a7, cbc-1=a3b-1 >
Subgroups: 242 in 83 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2.D8, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C32⋊4C8, C6×Dic3, C6×C12, C6.Q16, C3×C4⋊Dic3, C2×C32⋊4C8, C12.8Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D8, Q16, Dic6, C4×S3, C3⋊D4, C2.D8, S32, Dic3⋊C4, D4⋊S3, C3⋊Q16, C6.D6, D6⋊S3, C32⋊2Q8, C6.Q16, C32⋊2D8, C32⋊2Q16, C62.C22, C12.8Dic6
►On 96 points
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 25 68 95 9 29 64 87 5 33 72 91)(2 36 69 94 10 28 65 86 6 32 61 90)(3 35 70 93 11 27 66 85 7 31 62 89)(4 34 71 92 12 26 67 96 8 30 63 88)(13 45 77 55 17 41 81 51 21 37 73 59)(14 44 78 54 18 40 82 50 22 48 74 58)(15 43 79 53 19 39 83 49 23 47 75 57)(16 42 80 52 20 38 84 60 24 46 76 56)
(1 20 64 76)(2 15 65 83)(3 22 66 78)(4 17 67 73)(5 24 68 80)(6 19 69 75)(7 14 70 82)(8 21 71 77)(9 16 72 84)(10 23 61 79)(11 18 62 74)(12 13 63 81)(25 49 87 43)(26 56 88 38)(27 51 89 45)(28 58 90 40)(29 53 91 47)(30 60 92 42)(31 55 93 37)(32 50 94 44)(33 57 95 39)(34 52 96 46)(35 59 85 41)(36 54 86 48)
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,25,68,95,9,29,64,87,5,33,72,91)(2,36,69,94,10,28,65,86,6,32,61,90)(3,35,70,93,11,27,66,85,7,31,62,89)(4,34,71,92,12,26,67,96,8,30,63,88)(13,45,77,55,17,41,81,51,21,37,73,59)(14,44,78,54,18,40,82,50,22,48,74,58)(15,43,79,53,19,39,83,49,23,47,75,57)(16,42,80,52,20,38,84,60,24,46,76,56), (1,20,64,76)(2,15,65,83)(3,22,66,78)(4,17,67,73)(5,24,68,80)(6,19,69,75)(7,14,70,82)(8,21,71,77)(9,16,72,84)(10,23,61,79)(11,18,62,74)(12,13,63,81)(25,49,87,43)(26,56,88,38)(27,51,89,45)(28,58,90,40)(29,53,91,47)(30,60,92,42)(31,55,93,37)(32,50,94,44)(33,57,95,39)(34,52,96,46)(35,59,85,41)(36,54,86,48)>;
G:=Group( (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,25,68,95,9,29,64,87,5,33,72,91)(2,36,69,94,10,28,65,86,6,32,61,90)(3,35,70,93,11,27,66,85,7,31,62,89)(4,34,71,92,12,26,67,96,8,30,63,88)(13,45,77,55,17,41,81,51,21,37,73,59)(14,44,78,54,18,40,82,50,22,48,74,58)(15,43,79,53,19,39,83,49,23,47,75,57)(16,42,80,52,20,38,84,60,24,46,76,56), (1,20,64,76)(2,15,65,83)(3,22,66,78)(4,17,67,73)(5,24,68,80)(6,19,69,75)(7,14,70,82)(8,21,71,77)(9,16,72,84)(10,23,61,79)(11,18,62,74)(12,13,63,81)(25,49,87,43)(26,56,88,38)(27,51,89,45)(28,58,90,40)(29,53,91,47)(30,60,92,42)(31,55,93,37)(32,50,94,44)(33,57,95,39)(34,52,96,46)(35,59,85,41)(36,54,86,48) );
G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,25,68,95,9,29,64,87,5,33,72,91),(2,36,69,94,10,28,65,86,6,32,61,90),(3,35,70,93,11,27,66,85,7,31,62,89),(4,34,71,92,12,26,67,96,8,30,63,88),(13,45,77,55,17,41,81,51,21,37,73,59),(14,44,78,54,18,40,82,50,22,48,74,58),(15,43,79,53,19,39,83,49,23,47,75,57),(16,42,80,52,20,38,84,60,24,46,76,56)], [(1,20,64,76),(2,15,65,83),(3,22,66,78),(4,17,67,73),(5,24,68,80),(6,19,69,75),(7,14,70,82),(8,21,71,77),(9,16,72,84),(10,23,61,79),(11,18,62,74),(12,13,63,81),(25,49,87,43),(26,56,88,38),(27,51,89,45),(28,58,90,40),(29,53,91,47),(30,60,92,42),(31,55,93,37),(32,50,94,44),(33,57,95,39),(34,52,96,46),(35,59,85,41),(36,54,86,48)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | + | - | + | - | - | |||||
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | D8 | Q16 | Dic6 | C4×S3 | C3⋊D4 | S32 | D4⋊S3 | C3⋊Q16 | C6.D6 | C32⋊2Q8 | D6⋊S3 | C32⋊2D8 | C32⋊2Q16 |
| kernel | C12.8Dic6 | C3×C4⋊Dic3 | C2×C32⋊4C8 | C32⋊4C8 | C4⋊Dic3 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.8Dic6 ►in GL6(𝔽73)
| 1 | 71 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 21 | 25 | 0 | 0 | 0 | 0 |
| 70 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 30 |
| 0 | 0 | 0 | 0 | 43 | 43 |
| 35 | 61 | 0 | 0 | 0 | 0 |
| 29 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 68 |
| 0 | 0 | 0 | 0 | 18 | 50 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[21,70,0,0,0,0,25,52,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,13,43,0,0,0,0,30,43],[35,29,0,0,0,0,61,38,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,23,18,0,0,0,0,68,50] >;
C12.8Dic6 in GAP, Magma, Sage, TeX
C_{12}._8{\rm Dic}_6 % in TeX
G:=Group("C12.8Dic6"); // GroupNames label
G:=SmallGroup(288,224);
// by ID
G=gap.SmallGroup(288,224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b^-1>;
// generators/relations