metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.2D8, C4.10D24, C42.6D6, C12.22Q16, C12.1SD16, C4⋊C8.6S3, C4⋊Dic3.3C4, C4.7(C24⋊C2), (C2×C4).124D12, (C2×C12).466D4, C3⋊2(C4.10D8), C12⋊2Q8.9C2, C12⋊C8.10C2, (C4×C12).44C22, C4.10(D4.S3), C6.3(Q8⋊C4), C4.10(C3⋊Q16), C2.5(C2.D24), C6.13(D4⋊C4), C22.63(D6⋊C4), C2.5(C6.SD16), C6.4(C4.10D4), C2.5(C12.47D4), (C3×C4⋊C8).6C2, (C2×C4).17(C4×S3), (C2×C12).29(C2×C4), (C2×C4).230(C3⋊D4), (C2×C6).48(C22⋊C4), SmallGroup(192,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.2D8
G = < a,b,c | a12=b8=1, c2=a3, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >
Subgroups: 184 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4⋊C8, C4⋊C8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4.10D8, C12⋊C8, C3×C4⋊C8, C12⋊2Q8, C12.2D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, Q16, C4×S3, D12, C3⋊D4, C4.10D4, D4⋊C4, Q8⋊C4, C24⋊C2, D24, D6⋊C4, D4.S3, C3⋊Q16, C4.10D8, C6.SD16, C2.D24, C12.47D4, C12.2D8
►Regular action on 192 points
Generators in S192
(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)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 41 53 68 20 110 127 157)(2 48 54 63 21 117 128 164)(3 43 55 70 22 112 129 159)(4 38 56 65 23 119 130 166)(5 45 57 72 24 114 131 161)(6 40 58 67 13 109 132 168)(7 47 59 62 14 116 121 163)(8 42 60 69 15 111 122 158)(9 37 49 64 16 118 123 165)(10 44 50 71 17 113 124 160)(11 39 51 66 18 120 125 167)(12 46 52 61 19 115 126 162)(25 147 170 184 86 144 84 100)(26 154 171 191 87 139 73 107)(27 149 172 186 88 134 74 102)(28 156 173 181 89 141 75 97)(29 151 174 188 90 136 76 104)(30 146 175 183 91 143 77 99)(31 153 176 190 92 138 78 106)(32 148 177 185 93 133 79 101)(33 155 178 192 94 140 80 108)(34 150 179 187 95 135 81 103)(35 145 180 182 96 142 82 98)(36 152 169 189 85 137 83 105)
(1 80 4 83 7 74 10 77)(2 73 5 76 8 79 11 82)(3 78 6 81 9 84 12 75)(13 179 16 170 19 173 22 176)(14 172 17 175 20 178 23 169)(15 177 18 180 21 171 24 174)(25 126 28 129 31 132 34 123)(26 131 29 122 32 125 35 128)(27 124 30 127 33 130 36 121)(37 141 40 144 43 135 46 138)(38 134 41 137 44 140 47 143)(39 139 42 142 45 133 48 136)(49 86 52 89 55 92 58 95)(50 91 53 94 56 85 59 88)(51 96 54 87 57 90 60 93)(61 190 64 181 67 184 70 187)(62 183 65 186 68 189 71 192)(63 188 66 191 69 182 72 185)(97 168 100 159 103 162 106 165)(98 161 101 164 104 167 107 158)(99 166 102 157 105 160 108 163)(109 147 112 150 115 153 118 156)(110 152 113 155 116 146 119 149)(111 145 114 148 117 151 120 154)
G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,41,53,68,20,110,127,157)(2,48,54,63,21,117,128,164)(3,43,55,70,22,112,129,159)(4,38,56,65,23,119,130,166)(5,45,57,72,24,114,131,161)(6,40,58,67,13,109,132,168)(7,47,59,62,14,116,121,163)(8,42,60,69,15,111,122,158)(9,37,49,64,16,118,123,165)(10,44,50,71,17,113,124,160)(11,39,51,66,18,120,125,167)(12,46,52,61,19,115,126,162)(25,147,170,184,86,144,84,100)(26,154,171,191,87,139,73,107)(27,149,172,186,88,134,74,102)(28,156,173,181,89,141,75,97)(29,151,174,188,90,136,76,104)(30,146,175,183,91,143,77,99)(31,153,176,190,92,138,78,106)(32,148,177,185,93,133,79,101)(33,155,178,192,94,140,80,108)(34,150,179,187,95,135,81,103)(35,145,180,182,96,142,82,98)(36,152,169,189,85,137,83,105), (1,80,4,83,7,74,10,77)(2,73,5,76,8,79,11,82)(3,78,6,81,9,84,12,75)(13,179,16,170,19,173,22,176)(14,172,17,175,20,178,23,169)(15,177,18,180,21,171,24,174)(25,126,28,129,31,132,34,123)(26,131,29,122,32,125,35,128)(27,124,30,127,33,130,36,121)(37,141,40,144,43,135,46,138)(38,134,41,137,44,140,47,143)(39,139,42,142,45,133,48,136)(49,86,52,89,55,92,58,95)(50,91,53,94,56,85,59,88)(51,96,54,87,57,90,60,93)(61,190,64,181,67,184,70,187)(62,183,65,186,68,189,71,192)(63,188,66,191,69,182,72,185)(97,168,100,159,103,162,106,165)(98,161,101,164,104,167,107,158)(99,166,102,157,105,160,108,163)(109,147,112,150,115,153,118,156)(110,152,113,155,116,146,119,149)(111,145,114,148,117,151,120,154)>;
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)(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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,41,53,68,20,110,127,157)(2,48,54,63,21,117,128,164)(3,43,55,70,22,112,129,159)(4,38,56,65,23,119,130,166)(5,45,57,72,24,114,131,161)(6,40,58,67,13,109,132,168)(7,47,59,62,14,116,121,163)(8,42,60,69,15,111,122,158)(9,37,49,64,16,118,123,165)(10,44,50,71,17,113,124,160)(11,39,51,66,18,120,125,167)(12,46,52,61,19,115,126,162)(25,147,170,184,86,144,84,100)(26,154,171,191,87,139,73,107)(27,149,172,186,88,134,74,102)(28,156,173,181,89,141,75,97)(29,151,174,188,90,136,76,104)(30,146,175,183,91,143,77,99)(31,153,176,190,92,138,78,106)(32,148,177,185,93,133,79,101)(33,155,178,192,94,140,80,108)(34,150,179,187,95,135,81,103)(35,145,180,182,96,142,82,98)(36,152,169,189,85,137,83,105), (1,80,4,83,7,74,10,77)(2,73,5,76,8,79,11,82)(3,78,6,81,9,84,12,75)(13,179,16,170,19,173,22,176)(14,172,17,175,20,178,23,169)(15,177,18,180,21,171,24,174)(25,126,28,129,31,132,34,123)(26,131,29,122,32,125,35,128)(27,124,30,127,33,130,36,121)(37,141,40,144,43,135,46,138)(38,134,41,137,44,140,47,143)(39,139,42,142,45,133,48,136)(49,86,52,89,55,92,58,95)(50,91,53,94,56,85,59,88)(51,96,54,87,57,90,60,93)(61,190,64,181,67,184,70,187)(62,183,65,186,68,189,71,192)(63,188,66,191,69,182,72,185)(97,168,100,159,103,162,106,165)(98,161,101,164,104,167,107,158)(99,166,102,157,105,160,108,163)(109,147,112,150,115,153,118,156)(110,152,113,155,116,146,119,149)(111,145,114,148,117,151,120,154) );
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),(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),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,41,53,68,20,110,127,157),(2,48,54,63,21,117,128,164),(3,43,55,70,22,112,129,159),(4,38,56,65,23,119,130,166),(5,45,57,72,24,114,131,161),(6,40,58,67,13,109,132,168),(7,47,59,62,14,116,121,163),(8,42,60,69,15,111,122,158),(9,37,49,64,16,118,123,165),(10,44,50,71,17,113,124,160),(11,39,51,66,18,120,125,167),(12,46,52,61,19,115,126,162),(25,147,170,184,86,144,84,100),(26,154,171,191,87,139,73,107),(27,149,172,186,88,134,74,102),(28,156,173,181,89,141,75,97),(29,151,174,188,90,136,76,104),(30,146,175,183,91,143,77,99),(31,153,176,190,92,138,78,106),(32,148,177,185,93,133,79,101),(33,155,178,192,94,140,80,108),(34,150,179,187,95,135,81,103),(35,145,180,182,96,142,82,98),(36,152,169,189,85,137,83,105)], [(1,80,4,83,7,74,10,77),(2,73,5,76,8,79,11,82),(3,78,6,81,9,84,12,75),(13,179,16,170,19,173,22,176),(14,172,17,175,20,178,23,169),(15,177,18,180,21,171,24,174),(25,126,28,129,31,132,34,123),(26,131,29,122,32,125,35,128),(27,124,30,127,33,130,36,121),(37,141,40,144,43,135,46,138),(38,134,41,137,44,140,47,143),(39,139,42,142,45,133,48,136),(49,86,52,89,55,92,58,95),(50,91,53,94,56,85,59,88),(51,96,54,87,57,90,60,93),(61,190,64,181,67,184,70,187),(62,183,65,186,68,189,71,192),(63,188,66,191,69,182,72,185),(97,168,100,159,103,162,106,165),(98,161,101,164,104,167,107,158),(99,166,102,157,105,160,108,163),(109,147,112,150,115,153,118,156),(110,152,113,155,116,146,119,149),(111,145,114,148,117,151,120,154)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D8 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C24⋊C2 | D24 | C4.10D4 | D4.S3 | C3⋊Q16 | C12.47D4 |
| kernel | C12.2D8 | C12⋊C8 | C3×C4⋊C8 | C12⋊2Q8 | C4⋊Dic3 | C4⋊C8 | C2×C12 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 |
Matrix representation of C12.2D8 ►in GL4(𝔽73) generated by
| 65 | 0 | 0 | 0 |
| 44 | 9 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 72 | 1 |
| 63 | 0 | 0 | 0 |
| 27 | 22 | 0 | 0 |
| 0 | 0 | 17 | 57 |
| 0 | 0 | 9 | 56 |
| 69 | 25 | 0 | 0 |
| 11 | 4 | 0 | 0 |
| 0 | 0 | 41 | 32 |
| 0 | 0 | 57 | 0 |
G:=sub<GL(4,GF(73))| [65,44,0,0,0,9,0,0,0,0,72,72,0,0,2,1],[63,27,0,0,0,22,0,0,0,0,17,9,0,0,57,56],[69,11,0,0,25,4,0,0,0,0,41,57,0,0,32,0] >;
C12.2D8 in GAP, Magma, Sage, TeX
C_{12}._2D_8 % in TeX
G:=Group("C12.2D8"); // GroupNames label
G:=SmallGroup(192,45);
// by ID
G=gap.SmallGroup(192,45);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,141,36,1094,268,1123,346,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^3,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations