direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic31, C62⋊C4, C2.2D62, C22.D31, C62.4C22, C31⋊2(C2×C4), (C2×C62).C2, SmallGroup(248,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C2×Dic31 |
Generators and relations for C2×Dic31
G = < a,b,c | a2=b62=1, c2=b31, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic31
►Regular action on 248 points
Generators in S248
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 225)(164 226)(165 227)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 239)(178 240)(179 241)(180 242)(181 243)(182 244)(183 245)(184 246)(185 247)(186 248)
(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)
(1 125 32 156)(2 186 33 155)(3 185 34 154)(4 184 35 153)(5 183 36 152)(6 182 37 151)(7 181 38 150)(8 180 39 149)(9 179 40 148)(10 178 41 147)(11 177 42 146)(12 176 43 145)(13 175 44 144)(14 174 45 143)(15 173 46 142)(16 172 47 141)(17 171 48 140)(18 170 49 139)(19 169 50 138)(20 168 51 137)(21 167 52 136)(22 166 53 135)(23 165 54 134)(24 164 55 133)(25 163 56 132)(26 162 57 131)(27 161 58 130)(28 160 59 129)(29 159 60 128)(30 158 61 127)(31 157 62 126)(63 199 94 230)(64 198 95 229)(65 197 96 228)(66 196 97 227)(67 195 98 226)(68 194 99 225)(69 193 100 224)(70 192 101 223)(71 191 102 222)(72 190 103 221)(73 189 104 220)(74 188 105 219)(75 187 106 218)(76 248 107 217)(77 247 108 216)(78 246 109 215)(79 245 110 214)(80 244 111 213)(81 243 112 212)(82 242 113 211)(83 241 114 210)(84 240 115 209)(85 239 116 208)(86 238 117 207)(87 237 118 206)(88 236 119 205)(89 235 120 204)(90 234 121 203)(91 233 122 202)(92 232 123 201)(93 231 124 200)
G:=sub<Sym(248)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(183,245)(184,246)(185,247)(186,248), (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), (1,125,32,156)(2,186,33,155)(3,185,34,154)(4,184,35,153)(5,183,36,152)(6,182,37,151)(7,181,38,150)(8,180,39,149)(9,179,40,148)(10,178,41,147)(11,177,42,146)(12,176,43,145)(13,175,44,144)(14,174,45,143)(15,173,46,142)(16,172,47,141)(17,171,48,140)(18,170,49,139)(19,169,50,138)(20,168,51,137)(21,167,52,136)(22,166,53,135)(23,165,54,134)(24,164,55,133)(25,163,56,132)(26,162,57,131)(27,161,58,130)(28,160,59,129)(29,159,60,128)(30,158,61,127)(31,157,62,126)(63,199,94,230)(64,198,95,229)(65,197,96,228)(66,196,97,227)(67,195,98,226)(68,194,99,225)(69,193,100,224)(70,192,101,223)(71,191,102,222)(72,190,103,221)(73,189,104,220)(74,188,105,219)(75,187,106,218)(76,248,107,217)(77,247,108,216)(78,246,109,215)(79,245,110,214)(80,244,111,213)(81,243,112,212)(82,242,113,211)(83,241,114,210)(84,240,115,209)(85,239,116,208)(86,238,117,207)(87,237,118,206)(88,236,119,205)(89,235,120,204)(90,234,121,203)(91,233,122,202)(92,232,123,201)(93,231,124,200)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(183,245)(184,246)(185,247)(186,248), (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), (1,125,32,156)(2,186,33,155)(3,185,34,154)(4,184,35,153)(5,183,36,152)(6,182,37,151)(7,181,38,150)(8,180,39,149)(9,179,40,148)(10,178,41,147)(11,177,42,146)(12,176,43,145)(13,175,44,144)(14,174,45,143)(15,173,46,142)(16,172,47,141)(17,171,48,140)(18,170,49,139)(19,169,50,138)(20,168,51,137)(21,167,52,136)(22,166,53,135)(23,165,54,134)(24,164,55,133)(25,163,56,132)(26,162,57,131)(27,161,58,130)(28,160,59,129)(29,159,60,128)(30,158,61,127)(31,157,62,126)(63,199,94,230)(64,198,95,229)(65,197,96,228)(66,196,97,227)(67,195,98,226)(68,194,99,225)(69,193,100,224)(70,192,101,223)(71,191,102,222)(72,190,103,221)(73,189,104,220)(74,188,105,219)(75,187,106,218)(76,248,107,217)(77,247,108,216)(78,246,109,215)(79,245,110,214)(80,244,111,213)(81,243,112,212)(82,242,113,211)(83,241,114,210)(84,240,115,209)(85,239,116,208)(86,238,117,207)(87,237,118,206)(88,236,119,205)(89,235,120,204)(90,234,121,203)(91,233,122,202)(92,232,123,201)(93,231,124,200) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,197),(136,198),(137,199),(138,200),(139,201),(140,202),(141,203),(142,204),(143,205),(144,206),(145,207),(146,208),(147,209),(148,210),(149,211),(150,212),(151,213),(152,214),(153,215),(154,216),(155,217),(156,218),(157,219),(158,220),(159,221),(160,222),(161,223),(162,224),(163,225),(164,226),(165,227),(166,228),(167,229),(168,230),(169,231),(170,232),(171,233),(172,234),(173,235),(174,236),(175,237),(176,238),(177,239),(178,240),(179,241),(180,242),(181,243),(182,244),(183,245),(184,246),(185,247),(186,248)], [(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)], [(1,125,32,156),(2,186,33,155),(3,185,34,154),(4,184,35,153),(5,183,36,152),(6,182,37,151),(7,181,38,150),(8,180,39,149),(9,179,40,148),(10,178,41,147),(11,177,42,146),(12,176,43,145),(13,175,44,144),(14,174,45,143),(15,173,46,142),(16,172,47,141),(17,171,48,140),(18,170,49,139),(19,169,50,138),(20,168,51,137),(21,167,52,136),(22,166,53,135),(23,165,54,134),(24,164,55,133),(25,163,56,132),(26,162,57,131),(27,161,58,130),(28,160,59,129),(29,159,60,128),(30,158,61,127),(31,157,62,126),(63,199,94,230),(64,198,95,229),(65,197,96,228),(66,196,97,227),(67,195,98,226),(68,194,99,225),(69,193,100,224),(70,192,101,223),(71,191,102,222),(72,190,103,221),(73,189,104,220),(74,188,105,219),(75,187,106,218),(76,248,107,217),(77,247,108,216),(78,246,109,215),(79,245,110,214),(80,244,111,213),(81,243,112,212),(82,242,113,211),(83,241,114,210),(84,240,115,209),(85,239,116,208),(86,238,117,207),(87,237,118,206),(88,236,119,205),(89,235,120,204),(90,234,121,203),(91,233,122,202),(92,232,123,201),(93,231,124,200)]])
C2×Dic31 is a maximal subgroup of
Dic31⋊C4 C4⋊Dic31 D62⋊C4 C23.D31 C2×C4×D31 D4⋊2D31
C2×Dic31 is a maximal quotient of C4.Dic31 C4⋊Dic31 C23.D31
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 31A | ··· | 31O | 62A | ··· | 62AS |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D31 | Dic31 | D62 |
| kernel | C2×Dic31 | Dic31 | C2×C62 | C62 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 15 | 30 | 15 |
Matrix representation of C2×Dic31 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 372 |
| 0 | 1 | 300 |
| 372 | 0 | 0 |
| 0 | 149 | 95 |
| 0 | 155 | 224 |
G:=sub<GL(3,GF(373))| [372,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,372,300],[372,0,0,0,149,155,0,95,224] >;
C2×Dic31 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{31} % in TeX
G:=Group("C2xDic31"); // GroupNames label
G:=SmallGroup(248,6);
// by ID
G=gap.SmallGroup(248,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-31,16,3843]);
// Polycyclic
G:=Group<a,b,c|a^2=b^62=1,c^2=b^31,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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