metabelian, supersoluble, monomial
Aliases: C12.81D12, C12.14Dic6, Dic3⋊(C3⋊C8), C6.21(S3×C8), C32⋊6(C4⋊C8), (C3×Dic3)⋊1C8, C3⋊1(C12⋊C8), (C3×C12).17Q8, C3⋊2(Dic3⋊C8), (C3×C12).112D4, (C2×C12).297D6, C62.31(C2×C4), C6.1(C4⋊Dic3), (C4×Dic3).7S3, (C6×Dic3).5C4, C6.13(C8⋊S3), (C3×C6).7M4(2), C12.80(C3⋊D4), C6.1(Dic3⋊C4), (Dic3×C12).1C2, C4.7(C32⋊2Q8), C6.3(C4.Dic3), C4.30(C3⋊D12), (C6×C12).202C22, (C2×Dic3).4Dic3, C2.3(D6.Dic3), C22.11(S3×Dic3), C2.1(Dic3⋊Dic3), C6.5(C2×C3⋊C8), C2.5(S3×C3⋊C8), (C6×C3⋊C8).3C2, (C2×C3⋊C8).9S3, (C2×C4).130S32, (C2×C6).68(C4×S3), (C3×C6).20(C2×C8), (C3×C6).18(C4⋊C4), (C2×C6).15(C2×Dic3), (C2×C32⋊4C8).14C2, SmallGroup(288,219)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.81D12
G = < a,b,c | a12=b12=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >
Subgroups: 210 in 87 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4⋊C8, C3×Dic3, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, C32⋊4C8, C6×Dic3, C6×C12, C12⋊C8, Dic3⋊C8, C6×C3⋊C8, Dic3×C12, C2×C32⋊4C8, C12.81D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C2×C8, M4(2), C3⋊C8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C4⋊C8, S32, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C32⋊2Q8, C12⋊C8, Dic3⋊C8, S3×C3⋊C8, D6.Dic3, Dic3⋊Dic3, C12.81D12
►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 91 79 44 5 87 83 40 9 95 75 48)(2 96 80 37 6 92 84 45 10 88 76 41)(3 89 81 42 7 85 73 38 11 93 77 46)(4 94 82 47 8 90 74 43 12 86 78 39)(13 27 70 53 21 31 66 57 17 35 62 49)(14 32 71 58 22 36 67 50 18 28 63 54)(15 25 72 51 23 29 68 55 19 33 64 59)(16 30 61 56 24 34 69 60 20 26 65 52)
(1 50 10 59 7 56 4 53)(2 55 11 52 8 49 5 58)(3 60 12 57 9 54 6 51)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)(25 81 34 78 31 75 28 84)(26 74 35 83 32 80 29 77)(27 79 36 76 33 73 30 82)(61 94 70 91 67 88 64 85)(62 87 71 96 68 93 65 90)(63 92 72 89 69 86 66 95)
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), (1,91,79,44,5,87,83,40,9,95,75,48)(2,96,80,37,6,92,84,45,10,88,76,41)(3,89,81,42,7,85,73,38,11,93,77,46)(4,94,82,47,8,90,74,43,12,86,78,39)(13,27,70,53,21,31,66,57,17,35,62,49)(14,32,71,58,22,36,67,50,18,28,63,54)(15,25,72,51,23,29,68,55,19,33,64,59)(16,30,61,56,24,34,69,60,20,26,65,52), (1,50,10,59,7,56,4,53)(2,55,11,52,8,49,5,58)(3,60,12,57,9,54,6,51)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45)(25,81,34,78,31,75,28,84)(26,74,35,83,32,80,29,77)(27,79,36,76,33,73,30,82)(61,94,70,91,67,88,64,85)(62,87,71,96,68,93,65,90)(63,92,72,89,69,86,66,95)>;
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), (1,91,79,44,5,87,83,40,9,95,75,48)(2,96,80,37,6,92,84,45,10,88,76,41)(3,89,81,42,7,85,73,38,11,93,77,46)(4,94,82,47,8,90,74,43,12,86,78,39)(13,27,70,53,21,31,66,57,17,35,62,49)(14,32,71,58,22,36,67,50,18,28,63,54)(15,25,72,51,23,29,68,55,19,33,64,59)(16,30,61,56,24,34,69,60,20,26,65,52), (1,50,10,59,7,56,4,53)(2,55,11,52,8,49,5,58)(3,60,12,57,9,54,6,51)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45)(25,81,34,78,31,75,28,84)(26,74,35,83,32,80,29,77)(27,79,36,76,33,73,30,82)(61,94,70,91,67,88,64,85)(62,87,71,96,68,93,65,90)(63,92,72,89,69,86,66,95) );
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)], [(1,91,79,44,5,87,83,40,9,95,75,48),(2,96,80,37,6,92,84,45,10,88,76,41),(3,89,81,42,7,85,73,38,11,93,77,46),(4,94,82,47,8,90,74,43,12,86,78,39),(13,27,70,53,21,31,66,57,17,35,62,49),(14,32,71,58,22,36,67,50,18,28,63,54),(15,25,72,51,23,29,68,55,19,33,64,59),(16,30,61,56,24,34,69,60,20,26,65,52)], [(1,50,10,59,7,56,4,53),(2,55,11,52,8,49,5,58),(3,60,12,57,9,54,6,51),(13,44,22,41,19,38,16,47),(14,37,23,46,20,43,17,40),(15,42,24,39,21,48,18,45),(25,81,34,78,31,75,28,84),(26,74,35,83,32,80,29,77),(27,79,36,76,33,73,30,82),(61,94,70,91,67,88,64,85),(62,87,71,96,68,93,65,90),(63,92,72,89,69,86,66,95)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | S3 | D4 | Q8 | Dic3 | D6 | M4(2) | C3⋊C8 | Dic6 | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | C4.Dic3 | S32 | C3⋊D12 | C32⋊2Q8 | S3×Dic3 | S3×C3⋊C8 | D6.Dic3 |
| kernel | C12.81D12 | C6×C3⋊C8 | Dic3×C12 | C2×C32⋊4C8 | C6×Dic3 | C3×Dic3 | C2×C3⋊C8 | C4×Dic3 | C3×C12 | C3×C12 | C2×Dic3 | C2×C12 | C3×C6 | Dic3 | C12 | C12 | C12 | C2×C6 | C6 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C12.81D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 22 | 48 | 0 | 0 | 0 | 0 |
| 34 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 22 | 18 | 0 | 0 | 0 | 0 |
| 34 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,34,0,0,0,0,48,51,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,34,0,0,0,0,18,51,0,0,0,0,0,0,63,10,0,0,0,0,0,10,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
C12.81D12 in GAP, Magma, Sage, TeX
C_{12}._{81}D_{12} % in TeX
G:=Group("C12.81D12"); // GroupNames label
G:=SmallGroup(288,219);
// by ID
G=gap.SmallGroup(288,219);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations