metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊4Q16, C8.9D12, C4⋊1Dic12, C24.59D4, C42.266D6, (C4×C8).9S3, C6.4(C2×Q16), (C4×C24).11C2, (C2×C4).82D12, C4.33(C2×D12), C3⋊1(C4⋊Q16), (C2×C8).302D6, (C2×C12).379D4, C12.276(C2×D4), C6.6(C4⋊1D4), C12⋊2Q8.4C2, C2.6(C2×Dic12), C2.8(C4⋊D12), (C2×Dic12).2C2, C22.94(C2×D12), (C2×C12).728C23, (C2×C24).375C22, (C4×C12).311C22, (C2×Dic6).5C22, (C2×C6).111(C2×D4), (C2×C4).671(C22×S3), SmallGroup(192,258)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊4Q16
G = < a,b,c | a12=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 344 in 122 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C4×C8, C4⋊Q8, C2×Q16, Dic12, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4⋊Q16, C4×C24, C12⋊2Q8, C2×Dic12, C12⋊4Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D12, C22×S3, C4⋊1D4, C2×Q16, Dic12, C2×D12, C4⋊Q16, C4⋊D12, C2×Dic12, C12⋊4Q16
►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 25 19 151 188 102 76 179)(2 26 20 152 189 103 77 180)(3 27 21 153 190 104 78 169)(4 28 22 154 191 105 79 170)(5 29 23 155 192 106 80 171)(6 30 24 156 181 107 81 172)(7 31 13 145 182 108 82 173)(8 32 14 146 183 97 83 174)(9 33 15 147 184 98 84 175)(10 34 16 148 185 99 73 176)(11 35 17 149 186 100 74 177)(12 36 18 150 187 101 75 178)(37 55 70 139 121 162 87 116)(38 56 71 140 122 163 88 117)(39 57 72 141 123 164 89 118)(40 58 61 142 124 165 90 119)(41 59 62 143 125 166 91 120)(42 60 63 144 126 167 92 109)(43 49 64 133 127 168 93 110)(44 50 65 134 128 157 94 111)(45 51 66 135 129 158 95 112)(46 52 67 136 130 159 96 113)(47 53 68 137 131 160 85 114)(48 54 69 138 132 161 86 115)
(1 61 188 90)(2 72 189 89)(3 71 190 88)(4 70 191 87)(5 69 192 86)(6 68 181 85)(7 67 182 96)(8 66 183 95)(9 65 184 94)(10 64 185 93)(11 63 186 92)(12 62 187 91)(13 46 82 130)(14 45 83 129)(15 44 84 128)(16 43 73 127)(17 42 74 126)(18 41 75 125)(19 40 76 124)(20 39 77 123)(21 38 78 122)(22 37 79 121)(23 48 80 132)(24 47 81 131)(25 58 102 165)(26 57 103 164)(27 56 104 163)(28 55 105 162)(29 54 106 161)(30 53 107 160)(31 52 108 159)(32 51 97 158)(33 50 98 157)(34 49 99 168)(35 60 100 167)(36 59 101 166)(109 177 144 149)(110 176 133 148)(111 175 134 147)(112 174 135 146)(113 173 136 145)(114 172 137 156)(115 171 138 155)(116 170 139 154)(117 169 140 153)(118 180 141 152)(119 179 142 151)(120 178 143 150)
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), (1,25,19,151,188,102,76,179)(2,26,20,152,189,103,77,180)(3,27,21,153,190,104,78,169)(4,28,22,154,191,105,79,170)(5,29,23,155,192,106,80,171)(6,30,24,156,181,107,81,172)(7,31,13,145,182,108,82,173)(8,32,14,146,183,97,83,174)(9,33,15,147,184,98,84,175)(10,34,16,148,185,99,73,176)(11,35,17,149,186,100,74,177)(12,36,18,150,187,101,75,178)(37,55,70,139,121,162,87,116)(38,56,71,140,122,163,88,117)(39,57,72,141,123,164,89,118)(40,58,61,142,124,165,90,119)(41,59,62,143,125,166,91,120)(42,60,63,144,126,167,92,109)(43,49,64,133,127,168,93,110)(44,50,65,134,128,157,94,111)(45,51,66,135,129,158,95,112)(46,52,67,136,130,159,96,113)(47,53,68,137,131,160,85,114)(48,54,69,138,132,161,86,115), (1,61,188,90)(2,72,189,89)(3,71,190,88)(4,70,191,87)(5,69,192,86)(6,68,181,85)(7,67,182,96)(8,66,183,95)(9,65,184,94)(10,64,185,93)(11,63,186,92)(12,62,187,91)(13,46,82,130)(14,45,83,129)(15,44,84,128)(16,43,73,127)(17,42,74,126)(18,41,75,125)(19,40,76,124)(20,39,77,123)(21,38,78,122)(22,37,79,121)(23,48,80,132)(24,47,81,131)(25,58,102,165)(26,57,103,164)(27,56,104,163)(28,55,105,162)(29,54,106,161)(30,53,107,160)(31,52,108,159)(32,51,97,158)(33,50,98,157)(34,49,99,168)(35,60,100,167)(36,59,101,166)(109,177,144,149)(110,176,133,148)(111,175,134,147)(112,174,135,146)(113,173,136,145)(114,172,137,156)(115,171,138,155)(116,170,139,154)(117,169,140,153)(118,180,141,152)(119,179,142,151)(120,178,143,150)>;
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), (1,25,19,151,188,102,76,179)(2,26,20,152,189,103,77,180)(3,27,21,153,190,104,78,169)(4,28,22,154,191,105,79,170)(5,29,23,155,192,106,80,171)(6,30,24,156,181,107,81,172)(7,31,13,145,182,108,82,173)(8,32,14,146,183,97,83,174)(9,33,15,147,184,98,84,175)(10,34,16,148,185,99,73,176)(11,35,17,149,186,100,74,177)(12,36,18,150,187,101,75,178)(37,55,70,139,121,162,87,116)(38,56,71,140,122,163,88,117)(39,57,72,141,123,164,89,118)(40,58,61,142,124,165,90,119)(41,59,62,143,125,166,91,120)(42,60,63,144,126,167,92,109)(43,49,64,133,127,168,93,110)(44,50,65,134,128,157,94,111)(45,51,66,135,129,158,95,112)(46,52,67,136,130,159,96,113)(47,53,68,137,131,160,85,114)(48,54,69,138,132,161,86,115), (1,61,188,90)(2,72,189,89)(3,71,190,88)(4,70,191,87)(5,69,192,86)(6,68,181,85)(7,67,182,96)(8,66,183,95)(9,65,184,94)(10,64,185,93)(11,63,186,92)(12,62,187,91)(13,46,82,130)(14,45,83,129)(15,44,84,128)(16,43,73,127)(17,42,74,126)(18,41,75,125)(19,40,76,124)(20,39,77,123)(21,38,78,122)(22,37,79,121)(23,48,80,132)(24,47,81,131)(25,58,102,165)(26,57,103,164)(27,56,104,163)(28,55,105,162)(29,54,106,161)(30,53,107,160)(31,52,108,159)(32,51,97,158)(33,50,98,157)(34,49,99,168)(35,60,100,167)(36,59,101,166)(109,177,144,149)(110,176,133,148)(111,175,134,147)(112,174,135,146)(113,173,136,145)(114,172,137,156)(115,171,138,155)(116,170,139,154)(117,169,140,153)(118,180,141,152)(119,179,142,151)(120,178,143,150) );
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)], [(1,25,19,151,188,102,76,179),(2,26,20,152,189,103,77,180),(3,27,21,153,190,104,78,169),(4,28,22,154,191,105,79,170),(5,29,23,155,192,106,80,171),(6,30,24,156,181,107,81,172),(7,31,13,145,182,108,82,173),(8,32,14,146,183,97,83,174),(9,33,15,147,184,98,84,175),(10,34,16,148,185,99,73,176),(11,35,17,149,186,100,74,177),(12,36,18,150,187,101,75,178),(37,55,70,139,121,162,87,116),(38,56,71,140,122,163,88,117),(39,57,72,141,123,164,89,118),(40,58,61,142,124,165,90,119),(41,59,62,143,125,166,91,120),(42,60,63,144,126,167,92,109),(43,49,64,133,127,168,93,110),(44,50,65,134,128,157,94,111),(45,51,66,135,129,158,95,112),(46,52,67,136,130,159,96,113),(47,53,68,137,131,160,85,114),(48,54,69,138,132,161,86,115)], [(1,61,188,90),(2,72,189,89),(3,71,190,88),(4,70,191,87),(5,69,192,86),(6,68,181,85),(7,67,182,96),(8,66,183,95),(9,65,184,94),(10,64,185,93),(11,63,186,92),(12,62,187,91),(13,46,82,130),(14,45,83,129),(15,44,84,128),(16,43,73,127),(17,42,74,126),(18,41,75,125),(19,40,76,124),(20,39,77,123),(21,38,78,122),(22,37,79,121),(23,48,80,132),(24,47,81,131),(25,58,102,165),(26,57,103,164),(27,56,104,163),(28,55,105,162),(29,54,106,161),(30,53,107,160),(31,52,108,159),(32,51,97,158),(33,50,98,157),(34,49,99,168),(35,60,100,167),(36,59,101,166),(109,177,144,149),(110,176,133,148),(111,175,134,147),(112,174,135,146),(113,173,136,145),(114,172,137,156),(115,171,138,155),(116,170,139,154),(117,169,140,153),(118,180,141,152),(119,179,142,151),(120,178,143,150)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | D12 | D12 | Dic12 |
| kernel | C12⋊4Q16 | C4×C24 | C12⋊2Q8 | C2×Dic12 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C12⋊4Q16 ►in GL4(𝔽73) generated by
| 21 | 3 | 0 | 0 |
| 23 | 52 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 18 | 68 |
| 0 | 0 | 5 | 23 |
| 24 | 8 | 0 | 0 |
| 65 | 49 | 0 | 0 |
| 0 | 0 | 20 | 2 |
| 0 | 0 | 55 | 53 |
G:=sub<GL(4,GF(73))| [21,23,0,0,3,52,0,0,0,0,0,72,0,0,1,72],[72,0,0,0,0,72,0,0,0,0,18,5,0,0,68,23],[24,65,0,0,8,49,0,0,0,0,20,55,0,0,2,53] >;
C12⋊4Q16 in GAP, Magma, Sage, TeX
C_{12}\rtimes_4Q_{16} % in TeX
G:=Group("C12:4Q16"); // GroupNames label
G:=SmallGroup(192,258);
// by ID
G=gap.SmallGroup(192,258);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations