direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D31, C31⋊C24, C62⋊C23, (C22×C62)⋊3C2, (C2×C62)⋊4C22, SmallGroup(496,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C23×D31 |
Generators and relations for C23×D31
G = < a,b,c,d,e | a2=b2=c2=d31=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1664 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C24, C31, D31, C62, D62, C2×C62, C22×D31, C22×C62, C23×D31
Quotients: C1, C2, C22, C23, C24, D31, D62, C22×D31, C23×D31
►On 248 points
Generators in S248
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 125)(31 126)(32 184)(33 185)(34 186)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 181)(61 182)(62 183)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 196)(72 197)(73 198)(74 199)(75 200)(76 201)(77 202)(78 203)(79 204)(80 205)(81 206)(82 207)(83 208)(84 209)(85 210)(86 211)(87 212)(88 213)(89 214)(90 215)(91 216)(92 217)(93 187)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 236)(102 237)(103 238)(104 239)(105 240)(106 241)(107 242)(108 243)(109 244)(110 245)(111 246)(112 247)(113 248)(114 218)(115 219)(116 220)(117 221)(118 222)(119 223)(120 224)(121 225)(122 226)(123 227)(124 228)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(125 236)(126 237)(127 238)(128 239)(129 240)(130 241)(131 242)(132 243)(133 244)(134 245)(135 246)(136 247)(137 248)(138 218)(139 219)(140 220)(141 221)(142 222)(143 223)(144 224)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(151 231)(152 232)(153 233)(154 234)(155 235)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 211)(167 212)(168 213)(169 214)(170 215)(171 216)(172 217)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(183 197)(184 198)(185 199)(186 200)
(1 62)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 156)(132 157)(133 158)(134 159)(135 160)(136 161)(137 162)(138 163)(139 164)(140 165)(141 166)(142 167)(143 168)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(187 228)(188 229)(189 230)(190 231)(191 232)(192 233)(193 234)(194 235)(195 236)(196 237)(197 238)(198 239)(199 240)(200 241)(201 242)(202 243)(203 244)(204 245)(205 246)(206 247)(207 248)(208 218)(209 219)(210 220)(211 221)(212 222)(213 223)(214 224)(215 225)(216 226)(217 227)
(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)
(1 182)(2 181)(3 180)(4 179)(5 178)(6 177)(7 176)(8 175)(9 174)(10 173)(11 172)(12 171)(13 170)(14 169)(15 168)(16 167)(17 166)(18 165)(19 164)(20 163)(21 162)(22 161)(23 160)(24 159)(25 158)(26 157)(27 156)(28 186)(29 185)(30 184)(31 183)(32 125)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 126)(63 246)(64 245)(65 244)(66 243)(67 242)(68 241)(69 240)(70 239)(71 238)(72 237)(73 236)(74 235)(75 234)(76 233)(77 232)(78 231)(79 230)(80 229)(81 228)(82 227)(83 226)(84 225)(85 224)(86 223)(87 222)(88 221)(89 220)(90 219)(91 218)(92 248)(93 247)(94 205)(95 204)(96 203)(97 202)(98 201)(99 200)(100 199)(101 198)(102 197)(103 196)(104 195)(105 194)(106 193)(107 192)(108 191)(109 190)(110 189)(111 188)(112 187)(113 217)(114 216)(115 215)(116 214)(117 213)(118 212)(119 211)(120 210)(121 209)(122 208)(123 207)(124 206)
G:=sub<Sym(248)| (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,125)(31,126)(32,184)(33,185)(34,186)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,204)(80,205)(81,206)(82,207)(83,208)(84,209)(85,210)(86,211)(87,212)(88,213)(89,214)(90,215)(91,216)(92,217)(93,187)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,225)(122,226)(123,227)(124,228), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(125,236)(126,237)(127,238)(128,239)(129,240)(130,241)(131,242)(132,243)(133,244)(134,245)(135,246)(136,247)(137,248)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213)(169,214)(170,215)(171,216)(172,217)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(183,197)(184,198)(185,199)(186,200), (1,62)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,156)(132,157)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(141,166)(142,167)(143,168)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)(204,245)(205,246)(206,247)(207,248)(208,218)(209,219)(210,220)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227), (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), (1,182)(2,181)(3,180)(4,179)(5,178)(6,177)(7,176)(8,175)(9,174)(10,173)(11,172)(12,171)(13,170)(14,169)(15,168)(16,167)(17,166)(18,165)(19,164)(20,163)(21,162)(22,161)(23,160)(24,159)(25,158)(26,157)(27,156)(28,186)(29,185)(30,184)(31,183)(32,125)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,246)(64,245)(65,244)(66,243)(67,242)(68,241)(69,240)(70,239)(71,238)(72,237)(73,236)(74,235)(75,234)(76,233)(77,232)(78,231)(79,230)(80,229)(81,228)(82,227)(83,226)(84,225)(85,224)(86,223)(87,222)(88,221)(89,220)(90,219)(91,218)(92,248)(93,247)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,217)(114,216)(115,215)(116,214)(117,213)(118,212)(119,211)(120,210)(121,209)(122,208)(123,207)(124,206)>;
G:=Group( (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,125)(31,126)(32,184)(33,185)(34,186)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,204)(80,205)(81,206)(82,207)(83,208)(84,209)(85,210)(86,211)(87,212)(88,213)(89,214)(90,215)(91,216)(92,217)(93,187)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,225)(122,226)(123,227)(124,228), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(125,236)(126,237)(127,238)(128,239)(129,240)(130,241)(131,242)(132,243)(133,244)(134,245)(135,246)(136,247)(137,248)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213)(169,214)(170,215)(171,216)(172,217)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(183,197)(184,198)(185,199)(186,200), (1,62)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,156)(132,157)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(141,166)(142,167)(143,168)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)(204,245)(205,246)(206,247)(207,248)(208,218)(209,219)(210,220)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227), (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), (1,182)(2,181)(3,180)(4,179)(5,178)(6,177)(7,176)(8,175)(9,174)(10,173)(11,172)(12,171)(13,170)(14,169)(15,168)(16,167)(17,166)(18,165)(19,164)(20,163)(21,162)(22,161)(23,160)(24,159)(25,158)(26,157)(27,156)(28,186)(29,185)(30,184)(31,183)(32,125)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,246)(64,245)(65,244)(66,243)(67,242)(68,241)(69,240)(70,239)(71,238)(72,237)(73,236)(74,235)(75,234)(76,233)(77,232)(78,231)(79,230)(80,229)(81,228)(82,227)(83,226)(84,225)(85,224)(86,223)(87,222)(88,221)(89,220)(90,219)(91,218)(92,248)(93,247)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,217)(114,216)(115,215)(116,214)(117,213)(118,212)(119,211)(120,210)(121,209)(122,208)(123,207)(124,206) );
G=PermutationGroup([[(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,155),(30,125),(31,126),(32,184),(33,185),(34,186),(35,156),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,181),(61,182),(62,183),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,194),(70,195),(71,196),(72,197),(73,198),(74,199),(75,200),(76,201),(77,202),(78,203),(79,204),(80,205),(81,206),(82,207),(83,208),(84,209),(85,210),(86,211),(87,212),(88,213),(89,214),(90,215),(91,216),(92,217),(93,187),(94,229),(95,230),(96,231),(97,232),(98,233),(99,234),(100,235),(101,236),(102,237),(103,238),(104,239),(105,240),(106,241),(107,242),(108,243),(109,244),(110,245),(111,246),(112,247),(113,248),(114,218),(115,219),(116,220),(117,221),(118,222),(119,223),(120,224),(121,225),(122,226),(123,227),(124,228)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,115),(14,116),(15,117),(16,118),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(51,92),(52,93),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(125,236),(126,237),(127,238),(128,239),(129,240),(130,241),(131,242),(132,243),(133,244),(134,245),(135,246),(136,247),(137,248),(138,218),(139,219),(140,220),(141,221),(142,222),(143,223),(144,224),(145,225),(146,226),(147,227),(148,228),(149,229),(150,230),(151,231),(152,232),(153,233),(154,234),(155,235),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,211),(167,212),(168,213),(169,214),(170,215),(171,216),(172,217),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(183,197),(184,198),(185,199),(186,200)], [(1,62),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(125,181),(126,182),(127,183),(128,184),(129,185),(130,186),(131,156),(132,157),(133,158),(134,159),(135,160),(136,161),(137,162),(138,163),(139,164),(140,165),(141,166),(142,167),(143,168),(144,169),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,180),(187,228),(188,229),(189,230),(190,231),(191,232),(192,233),(193,234),(194,235),(195,236),(196,237),(197,238),(198,239),(199,240),(200,241),(201,242),(202,243),(203,244),(204,245),(205,246),(206,247),(207,248),(208,218),(209,219),(210,220),(211,221),(212,222),(213,223),(214,224),(215,225),(216,226),(217,227)], [(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)], [(1,182),(2,181),(3,180),(4,179),(5,178),(6,177),(7,176),(8,175),(9,174),(10,173),(11,172),(12,171),(13,170),(14,169),(15,168),(16,167),(17,166),(18,165),(19,164),(20,163),(21,162),(22,161),(23,160),(24,159),(25,158),(26,157),(27,156),(28,186),(29,185),(30,184),(31,183),(32,125),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,140),(49,139),(50,138),(51,137),(52,136),(53,135),(54,134),(55,133),(56,132),(57,131),(58,130),(59,129),(60,128),(61,127),(62,126),(63,246),(64,245),(65,244),(66,243),(67,242),(68,241),(69,240),(70,239),(71,238),(72,237),(73,236),(74,235),(75,234),(76,233),(77,232),(78,231),(79,230),(80,229),(81,228),(82,227),(83,226),(84,225),(85,224),(86,223),(87,222),(88,221),(89,220),(90,219),(91,218),(92,248),(93,247),(94,205),(95,204),(96,203),(97,202),(98,201),(99,200),(100,199),(101,198),(102,197),(103,196),(104,195),(105,194),(106,193),(107,192),(108,191),(109,190),(110,189),(111,188),(112,187),(113,217),(114,216),(115,215),(116,214),(117,213),(118,212),(119,211),(120,210),(121,209),(122,208),(123,207),(124,206)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 31A | ··· | 31O | 62A | ··· | 62DA |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | ··· | 1 | 31 | ··· | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D31 | D62 |
| kernel | C23×D31 | C22×D31 | C22×C62 | C23 | C22 |
| # reps | 1 | 14 | 1 | 15 | 105 |
Matrix representation of C23×D31 ►in GL5(𝔽311)
| 310 | 0 | 0 | 0 | 0 |
| 0 | 310 | 0 | 0 | 0 |
| 0 | 0 | 310 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 310 | 0 | 0 | 0 | 0 |
| 0 | 310 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 310 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 310 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 259 | 1 |
| 0 | 0 | 0 | 293 | 102 |
| 310 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 310 | 0 | 0 |
| 0 | 0 | 0 | 125 | 131 |
| 0 | 0 | 0 | 149 | 186 |
G:=sub<GL(5,GF(311))| [310,0,0,0,0,0,310,0,0,0,0,0,310,0,0,0,0,0,1,0,0,0,0,0,1],[310,0,0,0,0,0,310,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[310,0,0,0,0,0,1,0,0,0,0,0,310,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,259,293,0,0,0,1,102],[310,0,0,0,0,0,1,0,0,0,0,0,310,0,0,0,0,0,125,149,0,0,0,131,186] >;
C23×D31 in GAP, Magma, Sage, TeX
C_2^3\times D_{31} % in TeX
G:=Group("C2^3xD31"); // GroupNames label
G:=SmallGroup(496,41);
// by ID
G=gap.SmallGroup(496,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,12004]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^31=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations