direct product, abelian, monomial, 2-elementary
Aliases: C2×C124, SmallGroup(248,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C124 |
| C1 — C2×C124 |
| C1 — C2×C124 |
Generators and relations for C2×C124
G = < a,b | a2=b124=1, ab=ba >
Smallest permutation representation of C2×C124
►Regular action on 248 points
Generators in S248
(1 211)(2 212)(3 213)(4 214)(5 215)(6 216)(7 217)(8 218)(9 219)(10 220)(11 221)(12 222)(13 223)(14 224)(15 225)(16 226)(17 227)(18 228)(19 229)(20 230)(21 231)(22 232)(23 233)(24 234)(25 235)(26 236)(27 237)(28 238)(29 239)(30 240)(31 241)(32 242)(33 243)(34 244)(35 245)(36 246)(37 247)(38 248)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 169)(84 170)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 177)(92 178)(93 179)(94 180)(95 181)(96 182)(97 183)(98 184)(99 185)(100 186)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 197)(112 198)(113 199)(114 200)(115 201)(116 202)(117 203)(118 204)(119 205)(120 206)(121 207)(122 208)(123 209)(124 210)
(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)(125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
G:=sub<Sym(248)| (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,225)(16,226)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,241)(32,242)(33,243)(34,244)(35,245)(36,246)(37,247)(38,248)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210), (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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)>;
G:=Group( (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,225)(16,226)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,241)(32,242)(33,243)(34,244)(35,245)(36,246)(37,247)(38,248)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210), (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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248) );
G=PermutationGroup([[(1,211),(2,212),(3,213),(4,214),(5,215),(6,216),(7,217),(8,218),(9,219),(10,220),(11,221),(12,222),(13,223),(14,224),(15,225),(16,226),(17,227),(18,228),(19,229),(20,230),(21,231),(22,232),(23,233),(24,234),(25,235),(26,236),(27,237),(28,238),(29,239),(30,240),(31,241),(32,242),(33,243),(34,244),(35,245),(36,246),(37,247),(38,248),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(73,159),(74,160),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,167),(82,168),(83,169),(84,170),(85,171),(86,172),(87,173),(88,174),(89,175),(90,176),(91,177),(92,178),(93,179),(94,180),(95,181),(96,182),(97,183),(98,184),(99,185),(100,186),(101,187),(102,188),(103,189),(104,190),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,197),(112,198),(113,199),(114,200),(115,201),(116,202),(117,203),(118,204),(119,205),(120,206),(121,207),(122,208),(123,209),(124,210)], [(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),(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)]])
C2×C124 is a maximal subgroup of
C4.Dic31 Dic31⋊C4 C4⋊Dic31 D62⋊C4 D124⋊5C2
248 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 31A | ··· | 31AD | 62A | ··· | 62CL | 124A | ··· | 124DP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
248 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C31 | C62 | C62 | C124 |
| kernel | C2×C124 | C124 | C2×C62 | C62 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 30 | 60 | 30 | 120 |
Matrix representation of C2×C124 ►in GL2(𝔽373) generated by
| 372 | 0 |
| 0 | 372 |
| 254 | 0 |
| 0 | 142 |
G:=sub<GL(2,GF(373))| [372,0,0,372],[254,0,0,142] >;
C2×C124 in GAP, Magma, Sage, TeX
C_2\times C_{124} % in TeX
G:=Group("C2xC124"); // GroupNames label
G:=SmallGroup(248,8);
// by ID
G=gap.SmallGroup(248,8);
# by ID
G:=PCGroup([4,-2,-2,-31,-2,496]);
// Polycyclic
G:=Group<a,b|a^2=b^124=1,a*b=b*a>;
// generators/relations
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