metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.10D8, C12.6Q16, C42.11D6, C12.10SD16, C4⋊Q8.2S3, C4⋊C4.2Dic3, C4.13(D4⋊S3), (C2×C12).108D4, C3⋊3(C4.10D8), C4.7(D4.S3), C12⋊C8.13C2, C4.7(C3⋊Q16), (C4×C12).49C22, C4.9(Q8⋊2S3), C6.25(D4⋊C4), C6.13(Q8⋊C4), C6.9(C4.10D4), C2.5(Q8⋊2Dic3), C2.5(D4⋊Dic3), C2.4(C12.10D4), C22.43(C6.D4), (C3×C4⋊C4).2C4, (C3×C4⋊Q8).2C2, (C2×C12).173(C2×C4), (C2×C4).13(C2×Dic3), (C2×C4).178(C3⋊D4), (C2×C6).105(C22⋊C4), SmallGroup(192,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.10D8
G = < a,b,c | a12=b8=1, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b-1 >
Subgroups: 144 in 64 conjugacy classes, 35 normal (31 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C2×C12, C2×C12, C3×Q8, C4⋊C8, C4⋊Q8, C2×C3⋊C8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, C4.10D8, C12⋊C8, C3×C4⋊Q8, C12.10D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, Q16, C2×Dic3, C3⋊D4, C4.10D4, D4⋊C4, Q8⋊C4, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C6.D4, C4.10D8, D4⋊Dic3, Q8⋊2Dic3, C12.10D4, C12.10D8
►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 82 99 30 115 147 55 93)(2 81 100 29 116 146 56 92)(3 80 101 28 117 145 57 91)(4 79 102 27 118 156 58 90)(5 78 103 26 119 155 59 89)(6 77 104 25 120 154 60 88)(7 76 105 36 109 153 49 87)(8 75 106 35 110 152 50 86)(9 74 107 34 111 151 51 85)(10 73 108 33 112 150 52 96)(11 84 97 32 113 149 53 95)(12 83 98 31 114 148 54 94)(13 157 46 139 191 174 72 127)(14 168 47 138 192 173 61 126)(15 167 48 137 181 172 62 125)(16 166 37 136 182 171 63 124)(17 165 38 135 183 170 64 123)(18 164 39 134 184 169 65 122)(19 163 40 133 185 180 66 121)(20 162 41 144 186 179 67 132)(21 161 42 143 187 178 68 131)(22 160 43 142 188 177 69 130)(23 159 44 141 189 176 70 129)(24 158 45 140 190 175 71 128)
(1 188 10 185 7 182 4 191)(2 181 11 190 8 187 5 184)(3 186 12 183 9 192 6 189)(13 115 22 112 19 109 16 118)(14 120 23 117 20 114 17 111)(15 113 24 110 21 119 18 116)(25 162 34 159 31 168 28 165)(26 167 35 164 32 161 29 158)(27 160 36 157 33 166 30 163)(37 102 46 99 43 108 40 105)(38 107 47 104 44 101 41 98)(39 100 48 97 45 106 42 103)(49 63 58 72 55 69 52 66)(50 68 59 65 56 62 53 71)(51 61 60 70 57 67 54 64)(73 136 82 133 79 142 76 139)(74 141 83 138 80 135 77 144)(75 134 84 143 81 140 78 137)(85 176 94 173 91 170 88 179)(86 169 95 178 92 175 89 172)(87 174 96 171 93 180 90 177)(121 156 130 153 127 150 124 147)(122 149 131 146 128 155 125 152)(123 154 132 151 129 148 126 145)
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), (1,82,99,30,115,147,55,93)(2,81,100,29,116,146,56,92)(3,80,101,28,117,145,57,91)(4,79,102,27,118,156,58,90)(5,78,103,26,119,155,59,89)(6,77,104,25,120,154,60,88)(7,76,105,36,109,153,49,87)(8,75,106,35,110,152,50,86)(9,74,107,34,111,151,51,85)(10,73,108,33,112,150,52,96)(11,84,97,32,113,149,53,95)(12,83,98,31,114,148,54,94)(13,157,46,139,191,174,72,127)(14,168,47,138,192,173,61,126)(15,167,48,137,181,172,62,125)(16,166,37,136,182,171,63,124)(17,165,38,135,183,170,64,123)(18,164,39,134,184,169,65,122)(19,163,40,133,185,180,66,121)(20,162,41,144,186,179,67,132)(21,161,42,143,187,178,68,131)(22,160,43,142,188,177,69,130)(23,159,44,141,189,176,70,129)(24,158,45,140,190,175,71,128), (1,188,10,185,7,182,4,191)(2,181,11,190,8,187,5,184)(3,186,12,183,9,192,6,189)(13,115,22,112,19,109,16,118)(14,120,23,117,20,114,17,111)(15,113,24,110,21,119,18,116)(25,162,34,159,31,168,28,165)(26,167,35,164,32,161,29,158)(27,160,36,157,33,166,30,163)(37,102,46,99,43,108,40,105)(38,107,47,104,44,101,41,98)(39,100,48,97,45,106,42,103)(49,63,58,72,55,69,52,66)(50,68,59,65,56,62,53,71)(51,61,60,70,57,67,54,64)(73,136,82,133,79,142,76,139)(74,141,83,138,80,135,77,144)(75,134,84,143,81,140,78,137)(85,176,94,173,91,170,88,179)(86,169,95,178,92,175,89,172)(87,174,96,171,93,180,90,177)(121,156,130,153,127,150,124,147)(122,149,131,146,128,155,125,152)(123,154,132,151,129,148,126,145)>;
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), (1,82,99,30,115,147,55,93)(2,81,100,29,116,146,56,92)(3,80,101,28,117,145,57,91)(4,79,102,27,118,156,58,90)(5,78,103,26,119,155,59,89)(6,77,104,25,120,154,60,88)(7,76,105,36,109,153,49,87)(8,75,106,35,110,152,50,86)(9,74,107,34,111,151,51,85)(10,73,108,33,112,150,52,96)(11,84,97,32,113,149,53,95)(12,83,98,31,114,148,54,94)(13,157,46,139,191,174,72,127)(14,168,47,138,192,173,61,126)(15,167,48,137,181,172,62,125)(16,166,37,136,182,171,63,124)(17,165,38,135,183,170,64,123)(18,164,39,134,184,169,65,122)(19,163,40,133,185,180,66,121)(20,162,41,144,186,179,67,132)(21,161,42,143,187,178,68,131)(22,160,43,142,188,177,69,130)(23,159,44,141,189,176,70,129)(24,158,45,140,190,175,71,128), (1,188,10,185,7,182,4,191)(2,181,11,190,8,187,5,184)(3,186,12,183,9,192,6,189)(13,115,22,112,19,109,16,118)(14,120,23,117,20,114,17,111)(15,113,24,110,21,119,18,116)(25,162,34,159,31,168,28,165)(26,167,35,164,32,161,29,158)(27,160,36,157,33,166,30,163)(37,102,46,99,43,108,40,105)(38,107,47,104,44,101,41,98)(39,100,48,97,45,106,42,103)(49,63,58,72,55,69,52,66)(50,68,59,65,56,62,53,71)(51,61,60,70,57,67,54,64)(73,136,82,133,79,142,76,139)(74,141,83,138,80,135,77,144)(75,134,84,143,81,140,78,137)(85,176,94,173,91,170,88,179)(86,169,95,178,92,175,89,172)(87,174,96,171,93,180,90,177)(121,156,130,153,127,150,124,147)(122,149,131,146,128,155,125,152)(123,154,132,151,129,148,126,145) );
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)], [(1,82,99,30,115,147,55,93),(2,81,100,29,116,146,56,92),(3,80,101,28,117,145,57,91),(4,79,102,27,118,156,58,90),(5,78,103,26,119,155,59,89),(6,77,104,25,120,154,60,88),(7,76,105,36,109,153,49,87),(8,75,106,35,110,152,50,86),(9,74,107,34,111,151,51,85),(10,73,108,33,112,150,52,96),(11,84,97,32,113,149,53,95),(12,83,98,31,114,148,54,94),(13,157,46,139,191,174,72,127),(14,168,47,138,192,173,61,126),(15,167,48,137,181,172,62,125),(16,166,37,136,182,171,63,124),(17,165,38,135,183,170,64,123),(18,164,39,134,184,169,65,122),(19,163,40,133,185,180,66,121),(20,162,41,144,186,179,67,132),(21,161,42,143,187,178,68,131),(22,160,43,142,188,177,69,130),(23,159,44,141,189,176,70,129),(24,158,45,140,190,175,71,128)], [(1,188,10,185,7,182,4,191),(2,181,11,190,8,187,5,184),(3,186,12,183,9,192,6,189),(13,115,22,112,19,109,16,118),(14,120,23,117,20,114,17,111),(15,113,24,110,21,119,18,116),(25,162,34,159,31,168,28,165),(26,167,35,164,32,161,29,158),(27,160,36,157,33,166,30,163),(37,102,46,99,43,108,40,105),(38,107,47,104,44,101,41,98),(39,100,48,97,45,106,42,103),(49,63,58,72,55,69,52,66),(50,68,59,65,56,62,53,71),(51,61,60,70,57,67,54,64),(73,136,82,133,79,142,76,139),(74,141,83,138,80,135,77,144),(75,134,84,143,81,140,78,137),(85,176,94,173,91,170,88,179),(86,169,95,178,92,175,89,172),(87,174,96,171,93,180,90,177),(121,156,130,153,127,150,124,147),(122,149,131,146,128,155,125,152),(123,154,132,151,129,148,126,145)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D8 | SD16 | Q16 | C3⋊D4 | C4.10D4 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | C12.10D4 |
| kernel | C12.10D8 | C12⋊C8 | C3×C4⋊Q8 | C3×C4⋊C4 | C4⋊Q8 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of C12.10D8 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 3 | 0 | 0 |
| 0 | 0 | 25 | 56 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 43 | 13 | 0 | 0 | 0 | 0 |
| 43 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 55 | 0 | 0 |
| 0 | 0 | 60 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 42 |
| 0 | 0 | 0 | 0 | 33 | 35 |
| 43 | 13 | 0 | 0 | 0 | 0 |
| 43 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 48 | 0 | 0 |
| 0 | 0 | 35 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 62 |
| 0 | 0 | 0 | 0 | 30 | 16 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,17,25,0,0,0,0,3,56,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[43,43,0,0,0,0,13,30,0,0,0,0,0,0,38,60,0,0,0,0,55,35,0,0,0,0,0,0,50,33,0,0,0,0,42,35],[43,43,0,0,0,0,13,30,0,0,0,0,0,0,69,35,0,0,0,0,48,36,0,0,0,0,0,0,57,30,0,0,0,0,62,16] >;
C12.10D8 in GAP, Magma, Sage, TeX
C_{12}._{10}D_8 % in TeX
G:=Group("C12.10D8"); // GroupNames label
G:=SmallGroup(192,106);
// by ID
G=gap.SmallGroup(192,106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,120,219,100,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^-1>;
// generators/relations