direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic12, C12⋊5Q16, C42.265D6, C3⋊1(C4×Q16), (C4×C8).8S3, C6.8(C4×D4), C8.21(C4×S3), C6.3(C2×Q16), C24.51(C2×C4), (C4×C24).10C2, C6.6(C4○D8), (C2×C4).64D12, (C2×C8).289D6, C2.11(C4×D12), C2.3(C4○D24), (C2×C12).354D4, (C4×Dic6).3C2, C24⋊1C4.18C2, C2.1(C2×Dic12), Dic6.12(C2×C4), C22.29(C2×D12), C12.220(C4○D4), C4.104(C4○D12), (C2×C12).727C23, C12.103(C22×C4), (C4×C12).327C22, (C2×C24).349C22, (C2×Dic12).14C2, C2.Dic12.17C2, C4⋊Dic3.264C22, (C2×Dic6).207C22, C4.61(S3×C2×C4), (C2×C6).110(C2×D4), (C2×C4).670(C22×S3), SmallGroup(192,257)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic12
G = < a,b,c | a4=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 264 in 110 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C24, C24, Dic6, Dic6, C2×Dic3, C2×C12, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, Dic12, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4×Q16, C2.Dic12, C24⋊1C4, C4×C24, C4×Dic6, C2×Dic12, C4×Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, Q16, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×Q16, C4○D8, Dic12, S3×C2×C4, C2×D12, C4○D12, C4×Q16, C4×D12, C4○D24, C2×Dic12, C4×Dic12
►Regular action on 192 points
Generators in S192
(1 65 30 77)(2 66 31 78)(3 67 32 79)(4 68 33 80)(5 69 34 81)(6 70 35 82)(7 71 36 83)(8 72 37 84)(9 49 38 85)(10 50 39 86)(11 51 40 87)(12 52 41 88)(13 53 42 89)(14 54 43 90)(15 55 44 91)(16 56 45 92)(17 57 46 93)(18 58 47 94)(19 59 48 95)(20 60 25 96)(21 61 26 73)(22 62 27 74)(23 63 28 75)(24 64 29 76)(97 136 176 151)(98 137 177 152)(99 138 178 153)(100 139 179 154)(101 140 180 155)(102 141 181 156)(103 142 182 157)(104 143 183 158)(105 144 184 159)(106 121 185 160)(107 122 186 161)(108 123 187 162)(109 124 188 163)(110 125 189 164)(111 126 190 165)(112 127 191 166)(113 128 192 167)(114 129 169 168)(115 130 170 145)(116 131 171 146)(117 132 172 147)(118 133 173 148)(119 134 174 149)(120 135 175 150)
(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)
(1 185 13 173)(2 184 14 172)(3 183 15 171)(4 182 16 170)(5 181 17 169)(6 180 18 192)(7 179 19 191)(8 178 20 190)(9 177 21 189)(10 176 22 188)(11 175 23 187)(12 174 24 186)(25 111 37 99)(26 110 38 98)(27 109 39 97)(28 108 40 120)(29 107 41 119)(30 106 42 118)(31 105 43 117)(32 104 44 116)(33 103 45 115)(34 102 46 114)(35 101 47 113)(36 100 48 112)(49 152 61 164)(50 151 62 163)(51 150 63 162)(52 149 64 161)(53 148 65 160)(54 147 66 159)(55 146 67 158)(56 145 68 157)(57 168 69 156)(58 167 70 155)(59 166 71 154)(60 165 72 153)(73 125 85 137)(74 124 86 136)(75 123 87 135)(76 122 88 134)(77 121 89 133)(78 144 90 132)(79 143 91 131)(80 142 92 130)(81 141 93 129)(82 140 94 128)(83 139 95 127)(84 138 96 126)
G:=sub<Sym(192)| (1,65,30,77)(2,66,31,78)(3,67,32,79)(4,68,33,80)(5,69,34,81)(6,70,35,82)(7,71,36,83)(8,72,37,84)(9,49,38,85)(10,50,39,86)(11,51,40,87)(12,52,41,88)(13,53,42,89)(14,54,43,90)(15,55,44,91)(16,56,45,92)(17,57,46,93)(18,58,47,94)(19,59,48,95)(20,60,25,96)(21,61,26,73)(22,62,27,74)(23,63,28,75)(24,64,29,76)(97,136,176,151)(98,137,177,152)(99,138,178,153)(100,139,179,154)(101,140,180,155)(102,141,181,156)(103,142,182,157)(104,143,183,158)(105,144,184,159)(106,121,185,160)(107,122,186,161)(108,123,187,162)(109,124,188,163)(110,125,189,164)(111,126,190,165)(112,127,191,166)(113,128,192,167)(114,129,169,168)(115,130,170,145)(116,131,171,146)(117,132,172,147)(118,133,173,148)(119,134,174,149)(120,135,175,150), (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), (1,185,13,173)(2,184,14,172)(3,183,15,171)(4,182,16,170)(5,181,17,169)(6,180,18,192)(7,179,19,191)(8,178,20,190)(9,177,21,189)(10,176,22,188)(11,175,23,187)(12,174,24,186)(25,111,37,99)(26,110,38,98)(27,109,39,97)(28,108,40,120)(29,107,41,119)(30,106,42,118)(31,105,43,117)(32,104,44,116)(33,103,45,115)(34,102,46,114)(35,101,47,113)(36,100,48,112)(49,152,61,164)(50,151,62,163)(51,150,63,162)(52,149,64,161)(53,148,65,160)(54,147,66,159)(55,146,67,158)(56,145,68,157)(57,168,69,156)(58,167,70,155)(59,166,71,154)(60,165,72,153)(73,125,85,137)(74,124,86,136)(75,123,87,135)(76,122,88,134)(77,121,89,133)(78,144,90,132)(79,143,91,131)(80,142,92,130)(81,141,93,129)(82,140,94,128)(83,139,95,127)(84,138,96,126)>;
G:=Group( (1,65,30,77)(2,66,31,78)(3,67,32,79)(4,68,33,80)(5,69,34,81)(6,70,35,82)(7,71,36,83)(8,72,37,84)(9,49,38,85)(10,50,39,86)(11,51,40,87)(12,52,41,88)(13,53,42,89)(14,54,43,90)(15,55,44,91)(16,56,45,92)(17,57,46,93)(18,58,47,94)(19,59,48,95)(20,60,25,96)(21,61,26,73)(22,62,27,74)(23,63,28,75)(24,64,29,76)(97,136,176,151)(98,137,177,152)(99,138,178,153)(100,139,179,154)(101,140,180,155)(102,141,181,156)(103,142,182,157)(104,143,183,158)(105,144,184,159)(106,121,185,160)(107,122,186,161)(108,123,187,162)(109,124,188,163)(110,125,189,164)(111,126,190,165)(112,127,191,166)(113,128,192,167)(114,129,169,168)(115,130,170,145)(116,131,171,146)(117,132,172,147)(118,133,173,148)(119,134,174,149)(120,135,175,150), (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), (1,185,13,173)(2,184,14,172)(3,183,15,171)(4,182,16,170)(5,181,17,169)(6,180,18,192)(7,179,19,191)(8,178,20,190)(9,177,21,189)(10,176,22,188)(11,175,23,187)(12,174,24,186)(25,111,37,99)(26,110,38,98)(27,109,39,97)(28,108,40,120)(29,107,41,119)(30,106,42,118)(31,105,43,117)(32,104,44,116)(33,103,45,115)(34,102,46,114)(35,101,47,113)(36,100,48,112)(49,152,61,164)(50,151,62,163)(51,150,63,162)(52,149,64,161)(53,148,65,160)(54,147,66,159)(55,146,67,158)(56,145,68,157)(57,168,69,156)(58,167,70,155)(59,166,71,154)(60,165,72,153)(73,125,85,137)(74,124,86,136)(75,123,87,135)(76,122,88,134)(77,121,89,133)(78,144,90,132)(79,143,91,131)(80,142,92,130)(81,141,93,129)(82,140,94,128)(83,139,95,127)(84,138,96,126) );
G=PermutationGroup([[(1,65,30,77),(2,66,31,78),(3,67,32,79),(4,68,33,80),(5,69,34,81),(6,70,35,82),(7,71,36,83),(8,72,37,84),(9,49,38,85),(10,50,39,86),(11,51,40,87),(12,52,41,88),(13,53,42,89),(14,54,43,90),(15,55,44,91),(16,56,45,92),(17,57,46,93),(18,58,47,94),(19,59,48,95),(20,60,25,96),(21,61,26,73),(22,62,27,74),(23,63,28,75),(24,64,29,76),(97,136,176,151),(98,137,177,152),(99,138,178,153),(100,139,179,154),(101,140,180,155),(102,141,181,156),(103,142,182,157),(104,143,183,158),(105,144,184,159),(106,121,185,160),(107,122,186,161),(108,123,187,162),(109,124,188,163),(110,125,189,164),(111,126,190,165),(112,127,191,166),(113,128,192,167),(114,129,169,168),(115,130,170,145),(116,131,171,146),(117,132,172,147),(118,133,173,148),(119,134,174,149),(120,135,175,150)], [(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)], [(1,185,13,173),(2,184,14,172),(3,183,15,171),(4,182,16,170),(5,181,17,169),(6,180,18,192),(7,179,19,191),(8,178,20,190),(9,177,21,189),(10,176,22,188),(11,175,23,187),(12,174,24,186),(25,111,37,99),(26,110,38,98),(27,109,39,97),(28,108,40,120),(29,107,41,119),(30,106,42,118),(31,105,43,117),(32,104,44,116),(33,103,45,115),(34,102,46,114),(35,101,47,113),(36,100,48,112),(49,152,61,164),(50,151,62,163),(51,150,63,162),(52,149,64,161),(53,148,65,160),(54,147,66,159),(55,146,67,158),(56,145,68,157),(57,168,69,156),(58,167,70,155),(59,166,71,154),(60,165,72,153),(73,125,85,137),(74,124,86,136),(75,123,87,135),(76,122,88,134),(77,121,89,133),(78,144,90,132),(79,143,91,131),(80,142,92,130),(81,141,93,129),(82,140,94,128),(83,139,95,127),(84,138,96,126)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | Q16 | C4○D4 | C4×S3 | D12 | C4○D8 | Dic12 | C4○D12 | C4○D24 |
| kernel | C4×Dic12 | C2.Dic12 | C24⋊1C4 | C4×C24 | C4×Dic6 | C2×Dic12 | Dic12 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C4×Dic12 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 8 | 66 |
| 47 | 36 | 0 | 0 |
| 36 | 26 | 0 | 0 |
| 0 | 0 | 36 | 63 |
| 0 | 0 | 64 | 37 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[0,1,0,0,72,0,0,0,0,0,52,8,0,0,0,66],[47,36,0,0,36,26,0,0,0,0,36,64,0,0,63,37] >;
C4×Dic12 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{12} % in TeX
G:=Group("C4xDic12"); // GroupNames label
G:=SmallGroup(192,257);
// by ID
G=gap.SmallGroup(192,257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,58,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations