metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.53D8, C12.24Q16, C42.190D6, C12.1M4(2), C3⋊C8⋊1C8, C4⋊C8.1S3, C3⋊2(C8⋊1C8), C6.6(C4⋊C8), C4.11(S3×C8), C12.1(C2×C8), (C2×C12).31Q8, C12⋊C8.6C2, C6.1(C2.D8), C4.7(C8⋊S3), C4.26(D4⋊S3), (C2×C12).485D4, (C2×C4).18Dic6, C6.4(C8.C4), C2.3(Dic3⋊C8), (C4×C12).37C22, C4.12(C3⋊Q16), C2.1(C6.Q16), C2.1(C12.53D4), C22.18(Dic3⋊C4), (C4×C3⋊C8).1C2, (C2×C3⋊C8).4C4, (C3×C4⋊C8).1C2, (C2×C6).31(C4⋊C4), (C2×C12).46(C2×C4), (C2×C4).133(C4×S3), (C2×C4).263(C3⋊D4), SmallGroup(192,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.53D8
G = < a,b,c | a12=b8=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >
Smallest permutation representation of C12.53D8
►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 93 132 84 67 54 45 32)(2 86 121 77 68 59 46 25)(3 91 122 82 69 52 47 30)(4 96 123 75 70 57 48 35)(5 89 124 80 71 50 37 28)(6 94 125 73 72 55 38 33)(7 87 126 78 61 60 39 26)(8 92 127 83 62 53 40 31)(9 85 128 76 63 58 41 36)(10 90 129 81 64 51 42 29)(11 95 130 74 65 56 43 34)(12 88 131 79 66 49 44 27)(13 109 134 186 167 146 102 175)(14 114 135 191 168 151 103 180)(15 119 136 184 157 156 104 173)(16 112 137 189 158 149 105 178)(17 117 138 182 159 154 106 171)(18 110 139 187 160 147 107 176)(19 115 140 192 161 152 108 169)(20 120 141 185 162 145 97 174)(21 113 142 190 163 150 98 179)(22 118 143 183 164 155 99 172)(23 111 144 188 165 148 100 177)(24 116 133 181 166 153 101 170)
(1 13 10 22 7 19 4 16)(2 18 11 15 8 24 5 21)(3 23 12 20 9 17 6 14)(25 110 34 119 31 116 28 113)(26 115 35 112 32 109 29 118)(27 120 36 117 33 114 30 111)(37 142 46 139 43 136 40 133)(38 135 47 144 44 141 41 138)(39 140 48 137 45 134 42 143)(49 185 58 182 55 191 52 188)(50 190 59 187 56 184 53 181)(51 183 60 192 57 189 54 186)(61 161 70 158 67 167 64 164)(62 166 71 163 68 160 65 157)(63 159 72 168 69 165 66 162)(73 151 82 148 79 145 76 154)(74 156 83 153 80 150 77 147)(75 149 84 146 81 155 78 152)(85 171 94 180 91 177 88 174)(86 176 95 173 92 170 89 179)(87 169 96 178 93 175 90 172)(97 128 106 125 103 122 100 131)(98 121 107 130 104 127 101 124)(99 126 108 123 105 132 102 129)
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), (1,93,132,84,67,54,45,32)(2,86,121,77,68,59,46,25)(3,91,122,82,69,52,47,30)(4,96,123,75,70,57,48,35)(5,89,124,80,71,50,37,28)(6,94,125,73,72,55,38,33)(7,87,126,78,61,60,39,26)(8,92,127,83,62,53,40,31)(9,85,128,76,63,58,41,36)(10,90,129,81,64,51,42,29)(11,95,130,74,65,56,43,34)(12,88,131,79,66,49,44,27)(13,109,134,186,167,146,102,175)(14,114,135,191,168,151,103,180)(15,119,136,184,157,156,104,173)(16,112,137,189,158,149,105,178)(17,117,138,182,159,154,106,171)(18,110,139,187,160,147,107,176)(19,115,140,192,161,152,108,169)(20,120,141,185,162,145,97,174)(21,113,142,190,163,150,98,179)(22,118,143,183,164,155,99,172)(23,111,144,188,165,148,100,177)(24,116,133,181,166,153,101,170), (1,13,10,22,7,19,4,16)(2,18,11,15,8,24,5,21)(3,23,12,20,9,17,6,14)(25,110,34,119,31,116,28,113)(26,115,35,112,32,109,29,118)(27,120,36,117,33,114,30,111)(37,142,46,139,43,136,40,133)(38,135,47,144,44,141,41,138)(39,140,48,137,45,134,42,143)(49,185,58,182,55,191,52,188)(50,190,59,187,56,184,53,181)(51,183,60,192,57,189,54,186)(61,161,70,158,67,167,64,164)(62,166,71,163,68,160,65,157)(63,159,72,168,69,165,66,162)(73,151,82,148,79,145,76,154)(74,156,83,153,80,150,77,147)(75,149,84,146,81,155,78,152)(85,171,94,180,91,177,88,174)(86,176,95,173,92,170,89,179)(87,169,96,178,93,175,90,172)(97,128,106,125,103,122,100,131)(98,121,107,130,104,127,101,124)(99,126,108,123,105,132,102,129)>;
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), (1,93,132,84,67,54,45,32)(2,86,121,77,68,59,46,25)(3,91,122,82,69,52,47,30)(4,96,123,75,70,57,48,35)(5,89,124,80,71,50,37,28)(6,94,125,73,72,55,38,33)(7,87,126,78,61,60,39,26)(8,92,127,83,62,53,40,31)(9,85,128,76,63,58,41,36)(10,90,129,81,64,51,42,29)(11,95,130,74,65,56,43,34)(12,88,131,79,66,49,44,27)(13,109,134,186,167,146,102,175)(14,114,135,191,168,151,103,180)(15,119,136,184,157,156,104,173)(16,112,137,189,158,149,105,178)(17,117,138,182,159,154,106,171)(18,110,139,187,160,147,107,176)(19,115,140,192,161,152,108,169)(20,120,141,185,162,145,97,174)(21,113,142,190,163,150,98,179)(22,118,143,183,164,155,99,172)(23,111,144,188,165,148,100,177)(24,116,133,181,166,153,101,170), (1,13,10,22,7,19,4,16)(2,18,11,15,8,24,5,21)(3,23,12,20,9,17,6,14)(25,110,34,119,31,116,28,113)(26,115,35,112,32,109,29,118)(27,120,36,117,33,114,30,111)(37,142,46,139,43,136,40,133)(38,135,47,144,44,141,41,138)(39,140,48,137,45,134,42,143)(49,185,58,182,55,191,52,188)(50,190,59,187,56,184,53,181)(51,183,60,192,57,189,54,186)(61,161,70,158,67,167,64,164)(62,166,71,163,68,160,65,157)(63,159,72,168,69,165,66,162)(73,151,82,148,79,145,76,154)(74,156,83,153,80,150,77,147)(75,149,84,146,81,155,78,152)(85,171,94,180,91,177,88,174)(86,176,95,173,92,170,89,179)(87,169,96,178,93,175,90,172)(97,128,106,125,103,122,100,131)(98,121,107,130,104,127,101,124)(99,126,108,123,105,132,102,129) );
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)], [(1,93,132,84,67,54,45,32),(2,86,121,77,68,59,46,25),(3,91,122,82,69,52,47,30),(4,96,123,75,70,57,48,35),(5,89,124,80,71,50,37,28),(6,94,125,73,72,55,38,33),(7,87,126,78,61,60,39,26),(8,92,127,83,62,53,40,31),(9,85,128,76,63,58,41,36),(10,90,129,81,64,51,42,29),(11,95,130,74,65,56,43,34),(12,88,131,79,66,49,44,27),(13,109,134,186,167,146,102,175),(14,114,135,191,168,151,103,180),(15,119,136,184,157,156,104,173),(16,112,137,189,158,149,105,178),(17,117,138,182,159,154,106,171),(18,110,139,187,160,147,107,176),(19,115,140,192,161,152,108,169),(20,120,141,185,162,145,97,174),(21,113,142,190,163,150,98,179),(22,118,143,183,164,155,99,172),(23,111,144,188,165,148,100,177),(24,116,133,181,166,153,101,170)], [(1,13,10,22,7,19,4,16),(2,18,11,15,8,24,5,21),(3,23,12,20,9,17,6,14),(25,110,34,119,31,116,28,113),(26,115,35,112,32,109,29,118),(27,120,36,117,33,114,30,111),(37,142,46,139,43,136,40,133),(38,135,47,144,44,141,41,138),(39,140,48,137,45,134,42,143),(49,185,58,182,55,191,52,188),(50,190,59,187,56,184,53,181),(51,183,60,192,57,189,54,186),(61,161,70,158,67,167,64,164),(62,166,71,163,68,160,65,157),(63,159,72,168,69,165,66,162),(73,151,82,148,79,145,76,154),(74,156,83,153,80,150,77,147),(75,149,84,146,81,155,78,152),(85,171,94,180,91,177,88,174),(86,176,95,173,92,170,89,179),(87,169,96,178,93,175,90,172),(97,128,106,125,103,122,100,131),(98,121,107,130,104,127,101,124),(99,126,108,123,105,132,102,129)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 8M | 8N | 8O | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | D8 | Q16 | Dic6 | C4×S3 | C3⋊D4 | C8.C4 | S3×C8 | C8⋊S3 | D4⋊S3 | C3⋊Q16 | C12.53D4 |
| kernel | C12.53D8 | C4×C3⋊C8 | C12⋊C8 | C3×C4⋊C8 | C2×C3⋊C8 | C3⋊C8 | C4⋊C8 | C2×C12 | C2×C12 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of C12.53D8 ►in GL4(𝔽73) generated by
| 27 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 51 | 32 | 0 | 0 |
| 10 | 22 | 0 | 0 |
| 0 | 0 | 41 | 32 |
| 0 | 0 | 57 | 0 |
| 46 | 26 | 0 | 0 |
| 72 | 27 | 0 | 0 |
| 0 | 0 | 14 | 54 |
| 0 | 0 | 41 | 59 |
G:=sub<GL(4,GF(73))| [27,27,0,0,46,0,0,0,0,0,1,0,0,0,0,1],[51,10,0,0,32,22,0,0,0,0,41,57,0,0,32,0],[46,72,0,0,26,27,0,0,0,0,14,41,0,0,54,59] >;
C12.53D8 in GAP, Magma, Sage, TeX
C_{12}._{53}D_8 % in TeX
G:=Group("C12.53D8"); // GroupNames label
G:=SmallGroup(192,38);
// by ID
G=gap.SmallGroup(192,38);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations
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