direct product, abelian, monomial, 2-elementary
Aliases: C22×C62, SmallGroup(248,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C62 |
| C1 — C22×C62 |
| C1 — C22×C62 |
Generators and relations for C22×C62
G = < a,b,c | a2=b2=c62=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C62
►Regular action on 248 points
Generators in S248
(1 178)(2 179)(3 180)(4 181)(5 182)(6 183)(7 184)(8 185)(9 186)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 161)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 168)(54 169)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(101 232)(102 233)(103 234)(104 235)(105 236)(106 237)(107 238)(108 239)(109 240)(110 241)(111 242)(112 243)(113 244)(114 245)(115 246)(116 247)(117 248)(118 187)(119 188)(120 189)(121 190)(122 191)(123 192)(124 193)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(125 245)(126 246)(127 247)(128 248)(129 187)(130 188)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 197)(140 198)(141 199)(142 200)(143 201)(144 202)(145 203)(146 204)(147 205)(148 206)(149 207)(150 208)(151 209)(152 210)(153 211)(154 212)(155 213)(156 214)(157 215)(158 216)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 225)(168 226)(169 227)(170 228)(171 229)(172 230)(173 231)(174 232)(175 233)(176 234)(177 235)(178 236)(179 237)(180 238)(181 239)(182 240)(183 241)(184 242)(185 243)(186 244)
(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)| (1,178)(2,179)(3,180)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(101,232)(102,233)(103,234)(104,235)(105,236)(106,237)(107,238)(108,239)(109,240)(110,241)(111,242)(112,243)(113,244)(114,245)(115,246)(116,247)(117,248)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(125,245)(126,246)(127,247)(128,248)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,209)(152,210)(153,211)(154,212)(155,213)(156,214)(157,215)(158,216)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,233)(176,234)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,241)(184,242)(185,243)(186,244), (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( (1,178)(2,179)(3,180)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(101,232)(102,233)(103,234)(104,235)(105,236)(106,237)(107,238)(108,239)(109,240)(110,241)(111,242)(112,243)(113,244)(114,245)(115,246)(116,247)(117,248)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(125,245)(126,246)(127,247)(128,248)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,209)(152,210)(153,211)(154,212)(155,213)(156,214)(157,215)(158,216)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,233)(176,234)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,241)(184,242)(185,243)(186,244), (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([[(1,178),(2,179),(3,180),(4,181),(5,182),(6,183),(7,184),(8,185),(9,186),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,146),(32,147),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,156),(42,157),(43,158),(44,159),(45,160),(46,161),(47,162),(48,163),(49,164),(50,165),(51,166),(52,167),(53,168),(54,169),(55,170),(56,171),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,194),(64,195),(65,196),(66,197),(67,198),(68,199),(69,200),(70,201),(71,202),(72,203),(73,204),(74,205),(75,206),(76,207),(77,208),(78,209),(79,210),(80,211),(81,212),(82,213),(83,214),(84,215),(85,216),(86,217),(87,218),(88,219),(89,220),(90,221),(91,222),(92,223),(93,224),(94,225),(95,226),(96,227),(97,228),(98,229),(99,230),(100,231),(101,232),(102,233),(103,234),(104,235),(105,236),(106,237),(107,238),(108,239),(109,240),(110,241),(111,242),(112,243),(113,244),(114,245),(115,246),(116,247),(117,248),(118,187),(119,188),(120,189),(121,190),(122,191),(123,192),(124,193)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(125,245),(126,246),(127,247),(128,248),(129,187),(130,188),(131,189),(132,190),(133,191),(134,192),(135,193),(136,194),(137,195),(138,196),(139,197),(140,198),(141,199),(142,200),(143,201),(144,202),(145,203),(146,204),(147,205),(148,206),(149,207),(150,208),(151,209),(152,210),(153,211),(154,212),(155,213),(156,214),(157,215),(158,216),(159,217),(160,218),(161,219),(162,220),(163,221),(164,222),(165,223),(166,224),(167,225),(168,226),(169,227),(170,228),(171,229),(172,230),(173,231),(174,232),(175,233),(176,234),(177,235),(178,236),(179,237),(180,238),(181,239),(182,240),(183,241),(184,242),(185,243),(186,244)], [(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)]])
C22×C62 is a maximal subgroup of
C23.D31
248 conjugacy classes
| class | 1 | 2A | ··· | 2G | 31A | ··· | 31AD | 62A | ··· | 62HB |
| order | 1 | 2 | ··· | 2 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
248 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C31 | C62 |
| kernel | C22×C62 | C2×C62 | C23 | C22 |
| # reps | 1 | 7 | 30 | 210 |
Matrix representation of C22×C62 ►in GL3(𝔽311) generated by
| 1 | 0 | 0 |
| 0 | 310 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 310 |
| 61 | 0 | 0 |
| 0 | 83 | 0 |
| 0 | 0 | 225 |
G:=sub<GL(3,GF(311))| [1,0,0,0,310,0,0,0,1],[1,0,0,0,1,0,0,0,310],[61,0,0,0,83,0,0,0,225] >;
C22×C62 in GAP, Magma, Sage, TeX
C_2^2\times C_{62} % in TeX
G:=Group("C2^2xC62"); // GroupNames label
G:=SmallGroup(248,12);
// by ID
G=gap.SmallGroup(248,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-31]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^62=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export