metabelian, supersoluble, monomial
Aliases: C12.60D12, (C6×C24)⋊3C2, (C2×C24)⋊3S3, C3⋊2(D6⋊C8), C6.16(S3×C8), C6.25(D6⋊C4), (C3×C12).169D4, (C2×C12).420D6, C32⋊8(C22⋊C8), C62.74(C2×C4), C6.10(C8⋊S3), C2.3(C24⋊S3), C4.19(C12⋊S3), (C3×C6).17M4(2), C12.130(C3⋊D4), (C6×C12).342C22, C4.27(C32⋊7D4), C2.1(C6.11D12), (C2×C3⋊S3)⋊4C8, C2.5(C8×C3⋊S3), (C2×C8)⋊1(C3⋊S3), (C3×C6).36(C2×C8), (C2×C6).49(C4×S3), (C22×C3⋊S3).7C4, C22.11(C4×C3⋊S3), (C2×C32⋊4C8)⋊15C2, (C2×C3⋊Dic3).12C4, (C3×C6).56(C22⋊C4), (C2×C4×C3⋊S3).12C2, (C2×C4).93(C2×C3⋊S3), SmallGroup(288,295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.60D12
G = < a,b,c | a12=1, b12=a6, c2=a9, ab=ba, cac-1=a5, cbc-1=a3b11 >
Subgroups: 572 in 150 conjugacy classes, 61 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, C22×C4, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C2×C3⋊C8, C2×C24, S3×C2×C4, C32⋊4C8, C3×C24, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊C8, C2×C32⋊4C8, C6×C24, C2×C4×C3⋊S3, C12.60D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C3⋊S3, C4×S3, D12, C3⋊D4, C22⋊C8, C2×C3⋊S3, S3×C8, C8⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D6⋊C8, C8×C3⋊S3, C24⋊S3, C6.11D12, C12.60D12
►On 144 points
Generators in S144
(1 40 72 121 111 79 13 28 60 133 99 91)(2 41 49 122 112 80 14 29 61 134 100 92)(3 42 50 123 113 81 15 30 62 135 101 93)(4 43 51 124 114 82 16 31 63 136 102 94)(5 44 52 125 115 83 17 32 64 137 103 95)(6 45 53 126 116 84 18 33 65 138 104 96)(7 46 54 127 117 85 19 34 66 139 105 73)(8 47 55 128 118 86 20 35 67 140 106 74)(9 48 56 129 119 87 21 36 68 141 107 75)(10 25 57 130 120 88 22 37 69 142 108 76)(11 26 58 131 97 89 23 38 70 143 109 77)(12 27 59 132 98 90 24 39 71 144 110 78)
(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 132 133 12 13 144 121 24)(2 11 134 143 14 23 122 131)(3 142 135 22 15 130 123 10)(4 21 136 129 16 9 124 141)(5 128 137 8 17 140 125 20)(6 7 138 139 18 19 126 127)(25 81 108 50 37 93 120 62)(26 49 109 92 38 61 97 80)(27 91 110 60 39 79 98 72)(28 59 111 78 40 71 99 90)(29 77 112 70 41 89 100 58)(30 69 113 88 42 57 101 76)(31 87 114 56 43 75 102 68)(32 55 115 74 44 67 103 86)(33 73 116 66 45 85 104 54)(34 65 117 84 46 53 105 96)(35 83 118 52 47 95 106 64)(36 51 119 94 48 63 107 82)
G:=sub<Sym(144)| (1,40,72,121,111,79,13,28,60,133,99,91)(2,41,49,122,112,80,14,29,61,134,100,92)(3,42,50,123,113,81,15,30,62,135,101,93)(4,43,51,124,114,82,16,31,63,136,102,94)(5,44,52,125,115,83,17,32,64,137,103,95)(6,45,53,126,116,84,18,33,65,138,104,96)(7,46,54,127,117,85,19,34,66,139,105,73)(8,47,55,128,118,86,20,35,67,140,106,74)(9,48,56,129,119,87,21,36,68,141,107,75)(10,25,57,130,120,88,22,37,69,142,108,76)(11,26,58,131,97,89,23,38,70,143,109,77)(12,27,59,132,98,90,24,39,71,144,110,78), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,132,133,12,13,144,121,24)(2,11,134,143,14,23,122,131)(3,142,135,22,15,130,123,10)(4,21,136,129,16,9,124,141)(5,128,137,8,17,140,125,20)(6,7,138,139,18,19,126,127)(25,81,108,50,37,93,120,62)(26,49,109,92,38,61,97,80)(27,91,110,60,39,79,98,72)(28,59,111,78,40,71,99,90)(29,77,112,70,41,89,100,58)(30,69,113,88,42,57,101,76)(31,87,114,56,43,75,102,68)(32,55,115,74,44,67,103,86)(33,73,116,66,45,85,104,54)(34,65,117,84,46,53,105,96)(35,83,118,52,47,95,106,64)(36,51,119,94,48,63,107,82)>;
G:=Group( (1,40,72,121,111,79,13,28,60,133,99,91)(2,41,49,122,112,80,14,29,61,134,100,92)(3,42,50,123,113,81,15,30,62,135,101,93)(4,43,51,124,114,82,16,31,63,136,102,94)(5,44,52,125,115,83,17,32,64,137,103,95)(6,45,53,126,116,84,18,33,65,138,104,96)(7,46,54,127,117,85,19,34,66,139,105,73)(8,47,55,128,118,86,20,35,67,140,106,74)(9,48,56,129,119,87,21,36,68,141,107,75)(10,25,57,130,120,88,22,37,69,142,108,76)(11,26,58,131,97,89,23,38,70,143,109,77)(12,27,59,132,98,90,24,39,71,144,110,78), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,132,133,12,13,144,121,24)(2,11,134,143,14,23,122,131)(3,142,135,22,15,130,123,10)(4,21,136,129,16,9,124,141)(5,128,137,8,17,140,125,20)(6,7,138,139,18,19,126,127)(25,81,108,50,37,93,120,62)(26,49,109,92,38,61,97,80)(27,91,110,60,39,79,98,72)(28,59,111,78,40,71,99,90)(29,77,112,70,41,89,100,58)(30,69,113,88,42,57,101,76)(31,87,114,56,43,75,102,68)(32,55,115,74,44,67,103,86)(33,73,116,66,45,85,104,54)(34,65,117,84,46,53,105,96)(35,83,118,52,47,95,106,64)(36,51,119,94,48,63,107,82) );
G=PermutationGroup([[(1,40,72,121,111,79,13,28,60,133,99,91),(2,41,49,122,112,80,14,29,61,134,100,92),(3,42,50,123,113,81,15,30,62,135,101,93),(4,43,51,124,114,82,16,31,63,136,102,94),(5,44,52,125,115,83,17,32,64,137,103,95),(6,45,53,126,116,84,18,33,65,138,104,96),(7,46,54,127,117,85,19,34,66,139,105,73),(8,47,55,128,118,86,20,35,67,140,106,74),(9,48,56,129,119,87,21,36,68,141,107,75),(10,25,57,130,120,88,22,37,69,142,108,76),(11,26,58,131,97,89,23,38,70,143,109,77),(12,27,59,132,98,90,24,39,71,144,110,78)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,132,133,12,13,144,121,24),(2,11,134,143,14,23,122,131),(3,142,135,22,15,130,123,10),(4,21,136,129,16,9,124,141),(5,128,137,8,17,140,125,20),(6,7,138,139,18,19,126,127),(25,81,108,50,37,93,120,62),(26,49,109,92,38,61,97,80),(27,91,110,60,39,79,98,72),(28,59,111,78,40,71,99,90),(29,77,112,70,41,89,100,58),(30,69,113,88,42,57,101,76),(31,87,114,56,43,75,102,68),(32,55,115,74,44,67,103,86),(33,73,116,66,45,85,104,54),(34,65,117,84,46,53,105,96),(35,83,118,52,47,95,106,64),(36,51,119,94,48,63,107,82)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 |
| kernel | C12.60D12 | C2×C32⋊4C8 | C6×C24 | C2×C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C2×C3⋊S3 | C2×C24 | C3×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C12.60D12 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 43 | 30 | 0 | 0 |
| 43 | 13 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 63 | 10 |
| 43 | 30 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 10 | 63 |
| 0 | 0 | 0 | 63 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,27,0,0,0,0,27],[43,43,0,0,30,13,0,0,0,0,0,63,0,0,10,10],[43,60,0,0,30,30,0,0,0,0,10,0,0,0,63,63] >;
C12.60D12 in GAP, Magma, Sage, TeX
C_{12}._{60}D_{12} % in TeX
G:=Group("C12.60D12"); // GroupNames label
G:=SmallGroup(288,295);
// by ID
G=gap.SmallGroup(288,295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations