metabelian, supersoluble, monomial
Aliases: C12.6Dic6, C62.27D4, (C3×C12).6Q8, C12.21(C4×S3), C32⋊4C8⋊1C4, (C2×C12).81D6, C4⋊Dic3.9S3, C32⋊6(C4.Q8), C6.6(D4.S3), (C3×C6).14SD16, C6.4(Dic3⋊C4), (C6×C12).34C22, C6.6(Q8⋊2S3), C3⋊2(C12.Q8), C4.4(C32⋊2Q8), C4.6(C6.D6), C2.3(Dic6⋊S3), C22.9(D6⋊S3), C2.3(C62.C22), (C2×C4).102S32, (C3×C6).21(C4⋊C4), (C3×C12).32(C2×C4), (C3×C4⋊Dic3).4C2, (C2×C6).51(C3⋊D4), (C2×C32⋊4C8).2C2, SmallGroup(288,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.6Dic6
G = < a,b,c | a12=b12=1, c2=a6b6, bab-1=a-1, cac-1=a7, cbc-1=a9b-1 >
Subgroups: 242 in 83 conjugacy classes, 40 normal (14 characteristic)
C1, C2, 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×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C32⋊4C8, C6×Dic3, C6×C12, C12.Q8, C3×C4⋊Dic3, C2×C32⋊4C8, C12.6Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, SD16, Dic6, C4×S3, C3⋊D4, C4.Q8, S32, Dic3⋊C4, D4.S3, Q8⋊2S3, C6.D6, D6⋊S3, C32⋊2Q8, C12.Q8, Dic6⋊S3, C62.C22, C12.6Dic6
►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 42 95 36 9 46 91 28 5 38 87 32)(2 41 96 35 10 45 92 27 6 37 88 31)(3 40 85 34 11 44 93 26 7 48 89 30)(4 39 86 33 12 43 94 25 8 47 90 29)(13 52 63 80 17 60 67 76 21 56 71 84)(14 51 64 79 18 59 68 75 22 55 72 83)(15 50 65 78 19 58 69 74 23 54 61 82)(16 49 66 77 20 57 70 73 24 53 62 81)
(1 16 85 64)(2 23 86 71)(3 18 87 66)(4 13 88 61)(5 20 89 68)(6 15 90 63)(7 22 91 70)(8 17 92 65)(9 24 93 72)(10 19 94 67)(11 14 95 62)(12 21 96 69)(25 51 45 81)(26 58 46 76)(27 53 47 83)(28 60 48 78)(29 55 37 73)(30 50 38 80)(31 57 39 75)(32 52 40 82)(33 59 41 77)(34 54 42 84)(35 49 43 79)(36 56 44 74)
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), (1,42,95,36,9,46,91,28,5,38,87,32)(2,41,96,35,10,45,92,27,6,37,88,31)(3,40,85,34,11,44,93,26,7,48,89,30)(4,39,86,33,12,43,94,25,8,47,90,29)(13,52,63,80,17,60,67,76,21,56,71,84)(14,51,64,79,18,59,68,75,22,55,72,83)(15,50,65,78,19,58,69,74,23,54,61,82)(16,49,66,77,20,57,70,73,24,53,62,81), (1,16,85,64)(2,23,86,71)(3,18,87,66)(4,13,88,61)(5,20,89,68)(6,15,90,63)(7,22,91,70)(8,17,92,65)(9,24,93,72)(10,19,94,67)(11,14,95,62)(12,21,96,69)(25,51,45,81)(26,58,46,76)(27,53,47,83)(28,60,48,78)(29,55,37,73)(30,50,38,80)(31,57,39,75)(32,52,40,82)(33,59,41,77)(34,54,42,84)(35,49,43,79)(36,56,44,74)>;
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), (1,42,95,36,9,46,91,28,5,38,87,32)(2,41,96,35,10,45,92,27,6,37,88,31)(3,40,85,34,11,44,93,26,7,48,89,30)(4,39,86,33,12,43,94,25,8,47,90,29)(13,52,63,80,17,60,67,76,21,56,71,84)(14,51,64,79,18,59,68,75,22,55,72,83)(15,50,65,78,19,58,69,74,23,54,61,82)(16,49,66,77,20,57,70,73,24,53,62,81), (1,16,85,64)(2,23,86,71)(3,18,87,66)(4,13,88,61)(5,20,89,68)(6,15,90,63)(7,22,91,70)(8,17,92,65)(9,24,93,72)(10,19,94,67)(11,14,95,62)(12,21,96,69)(25,51,45,81)(26,58,46,76)(27,53,47,83)(28,60,48,78)(29,55,37,73)(30,50,38,80)(31,57,39,75)(32,52,40,82)(33,59,41,77)(34,54,42,84)(35,49,43,79)(36,56,44,74) );
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)], [(1,42,95,36,9,46,91,28,5,38,87,32),(2,41,96,35,10,45,92,27,6,37,88,31),(3,40,85,34,11,44,93,26,7,48,89,30),(4,39,86,33,12,43,94,25,8,47,90,29),(13,52,63,80,17,60,67,76,21,56,71,84),(14,51,64,79,18,59,68,75,22,55,72,83),(15,50,65,78,19,58,69,74,23,54,61,82),(16,49,66,77,20,57,70,73,24,53,62,81)], [(1,16,85,64),(2,23,86,71),(3,18,87,66),(4,13,88,61),(5,20,89,68),(6,15,90,63),(7,22,91,70),(8,17,92,65),(9,24,93,72),(10,19,94,67),(11,14,95,62),(12,21,96,69),(25,51,45,81),(26,58,46,76),(27,53,47,83),(28,60,48,78),(29,55,37,73),(30,50,38,80),(31,57,39,75),(32,52,40,82),(33,59,41,77),(34,54,42,84),(35,49,43,79),(36,56,44,74)]])
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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | - | + | + | - | - | |||||
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | SD16 | Dic6 | C4×S3 | C3⋊D4 | S32 | D4.S3 | Q8⋊2S3 | C6.D6 | C32⋊2Q8 | D6⋊S3 | Dic6⋊S3 |
| kernel | C12.6Dic6 | C3×C4⋊Dic3 | C2×C32⋊4C8 | C32⋊4C8 | C4⋊Dic3 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 1 | 1 | 4 |
Matrix representation of C12.6Dic6 ►in GL6(𝔽73)
| 46 | 3 | 0 | 0 | 0 | 0 |
| 0 | 27 | 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 |
| 58 | 3 | 0 | 0 | 0 | 0 |
| 22 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 43 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 35 | 21 | 0 | 0 | 0 | 0 |
| 46 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 2 | 62 |
G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,3,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[58,22,0,0,0,0,3,15,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,30,30,0,0,0,0,43,60],[35,46,0,0,0,0,21,38,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,2,0,0,0,0,13,62] >;
C12.6Dic6 in GAP, Magma, Sage, TeX
C_{12}._6{\rm Dic}_6 % in TeX
G:=Group("C12.6Dic6"); // GroupNames label
G:=SmallGroup(288,222);
// by ID
G=gap.SmallGroup(288,222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,148,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^6*b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b^-1>;
// generators/relations