metabelian, supersoluble, monomial
Aliases: C12.59D12, C24.3Dic3, C62.14Q8, (C3×C24).6C4, (C6×C24).9C2, (C2×C24).16S3, (C2×C12).397D6, (C3×C12).139D4, C8.1(C3⋊Dic3), (C2×C6).17Dic6, C3⋊2(C24.C4), C6.15(C4⋊Dic3), C32⋊8(C8.C4), C12.48(C2×Dic3), C4.18(C12⋊S3), (C6×C12).307C22, C12.58D6.1C2, C2.5(C12⋊Dic3), C22.2(C32⋊4Q8), (C2×C8).5(C3⋊S3), C4.8(C2×C3⋊Dic3), (C3×C6).42(C4⋊C4), (C3×C12).125(C2×C4), (C2×C4).74(C2×C3⋊S3), SmallGroup(288,294)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.59D12
G = < a,b,c | a12=1, b12=a6, c2=a3, ab=ba, cac-1=a5, cbc-1=b11 >
Subgroups: 220 in 90 conjugacy classes, 57 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C8, C2×C4, C32, C12, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C8.C4, C3×C12, C62, C4.Dic3, C2×C24, C32⋊4C8, C3×C24, C6×C12, C24.C4, C12.58D6, C6×C24, C12.59D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, D12, C2×Dic3, C8.C4, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C24.C4, C12⋊Dic3, C12.59D12
►On 144 points
Generators in S144
(1 75 35 19 93 29 13 87 47 7 81 41)(2 76 36 20 94 30 14 88 48 8 82 42)(3 77 37 21 95 31 15 89 25 9 83 43)(4 78 38 22 96 32 16 90 26 10 84 44)(5 79 39 23 73 33 17 91 27 11 85 45)(6 80 40 24 74 34 18 92 28 12 86 46)(49 109 126 55 115 132 61 97 138 67 103 144)(50 110 127 56 116 133 62 98 139 68 104 121)(51 111 128 57 117 134 63 99 140 69 105 122)(52 112 129 58 118 135 64 100 141 70 106 123)(53 113 130 59 119 136 65 101 142 71 107 124)(54 114 131 60 120 137 66 102 143 72 108 125)
(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)
(1 119 19 101 13 107 7 113)(2 106 20 112 14 118 8 100)(3 117 21 99 15 105 9 111)(4 104 22 110 16 116 10 98)(5 115 23 97 17 103 11 109)(6 102 24 108 18 114 12 120)(25 140 43 122 37 128 31 134)(26 127 44 133 38 139 32 121)(27 138 45 144 39 126 33 132)(28 125 46 131 40 137 34 143)(29 136 47 142 41 124 35 130)(30 123 48 129 42 135 36 141)(49 91 55 85 61 79 67 73)(50 78 56 96 62 90 68 84)(51 89 57 83 63 77 69 95)(52 76 58 94 64 88 70 82)(53 87 59 81 65 75 71 93)(54 74 60 92 66 86 72 80)
G:=sub<Sym(144)| (1,75,35,19,93,29,13,87,47,7,81,41)(2,76,36,20,94,30,14,88,48,8,82,42)(3,77,37,21,95,31,15,89,25,9,83,43)(4,78,38,22,96,32,16,90,26,10,84,44)(5,79,39,23,73,33,17,91,27,11,85,45)(6,80,40,24,74,34,18,92,28,12,86,46)(49,109,126,55,115,132,61,97,138,67,103,144)(50,110,127,56,116,133,62,98,139,68,104,121)(51,111,128,57,117,134,63,99,140,69,105,122)(52,112,129,58,118,135,64,100,141,70,106,123)(53,113,130,59,119,136,65,101,142,71,107,124)(54,114,131,60,120,137,66,102,143,72,108,125), (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), (1,119,19,101,13,107,7,113)(2,106,20,112,14,118,8,100)(3,117,21,99,15,105,9,111)(4,104,22,110,16,116,10,98)(5,115,23,97,17,103,11,109)(6,102,24,108,18,114,12,120)(25,140,43,122,37,128,31,134)(26,127,44,133,38,139,32,121)(27,138,45,144,39,126,33,132)(28,125,46,131,40,137,34,143)(29,136,47,142,41,124,35,130)(30,123,48,129,42,135,36,141)(49,91,55,85,61,79,67,73)(50,78,56,96,62,90,68,84)(51,89,57,83,63,77,69,95)(52,76,58,94,64,88,70,82)(53,87,59,81,65,75,71,93)(54,74,60,92,66,86,72,80)>;
G:=Group( (1,75,35,19,93,29,13,87,47,7,81,41)(2,76,36,20,94,30,14,88,48,8,82,42)(3,77,37,21,95,31,15,89,25,9,83,43)(4,78,38,22,96,32,16,90,26,10,84,44)(5,79,39,23,73,33,17,91,27,11,85,45)(6,80,40,24,74,34,18,92,28,12,86,46)(49,109,126,55,115,132,61,97,138,67,103,144)(50,110,127,56,116,133,62,98,139,68,104,121)(51,111,128,57,117,134,63,99,140,69,105,122)(52,112,129,58,118,135,64,100,141,70,106,123)(53,113,130,59,119,136,65,101,142,71,107,124)(54,114,131,60,120,137,66,102,143,72,108,125), (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), (1,119,19,101,13,107,7,113)(2,106,20,112,14,118,8,100)(3,117,21,99,15,105,9,111)(4,104,22,110,16,116,10,98)(5,115,23,97,17,103,11,109)(6,102,24,108,18,114,12,120)(25,140,43,122,37,128,31,134)(26,127,44,133,38,139,32,121)(27,138,45,144,39,126,33,132)(28,125,46,131,40,137,34,143)(29,136,47,142,41,124,35,130)(30,123,48,129,42,135,36,141)(49,91,55,85,61,79,67,73)(50,78,56,96,62,90,68,84)(51,89,57,83,63,77,69,95)(52,76,58,94,64,88,70,82)(53,87,59,81,65,75,71,93)(54,74,60,92,66,86,72,80) );
G=PermutationGroup([[(1,75,35,19,93,29,13,87,47,7,81,41),(2,76,36,20,94,30,14,88,48,8,82,42),(3,77,37,21,95,31,15,89,25,9,83,43),(4,78,38,22,96,32,16,90,26,10,84,44),(5,79,39,23,73,33,17,91,27,11,85,45),(6,80,40,24,74,34,18,92,28,12,86,46),(49,109,126,55,115,132,61,97,138,67,103,144),(50,110,127,56,116,133,62,98,139,68,104,121),(51,111,128,57,117,134,63,99,140,69,105,122),(52,112,129,58,118,135,64,100,141,70,106,123),(53,113,130,59,119,136,65,101,142,71,107,124),(54,114,131,60,120,137,66,102,143,72,108,125)], [(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)], [(1,119,19,101,13,107,7,113),(2,106,20,112,14,118,8,100),(3,117,21,99,15,105,9,111),(4,104,22,110,16,116,10,98),(5,115,23,97,17,103,11,109),(6,102,24,108,18,114,12,120),(25,140,43,122,37,128,31,134),(26,127,44,133,38,139,32,121),(27,138,45,144,39,126,33,132),(28,125,46,131,40,137,34,143),(29,136,47,142,41,124,35,130),(30,123,48,129,42,135,36,141),(49,91,55,85,61,79,67,73),(50,78,56,96,62,90,68,84),(51,89,57,83,63,77,69,95),(52,76,58,94,64,88,70,82),(53,87,59,81,65,75,71,93),(54,74,60,92,66,86,72,80)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D12 | Dic6 | C8.C4 | C24.C4 |
| kernel | C12.59D12 | C12.58D6 | C6×C24 | C3×C24 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 32 |
Matrix representation of C12.59D12 ►in GL4(𝔽73) generated by
| 24 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 49 |
| 17 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 30 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 17 | 0 |
G:=sub<GL(4,GF(73))| [24,0,0,0,0,3,0,0,0,0,70,0,0,0,0,49],[17,0,0,0,0,30,0,0,0,0,52,0,0,0,0,7],[0,52,0,0,30,0,0,0,0,0,0,17,0,0,7,0] >;
C12.59D12 in GAP, Magma, Sage, TeX
C_{12}._{59}D_{12} % in TeX
G:=Group("C12.59D12"); // GroupNames label
G:=SmallGroup(288,294);
// by ID
G=gap.SmallGroup(288,294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^11>;
// generators/relations