metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D124⋊5C2, C4.16D62, Dic62⋊5C2, C62.4C23, C22.2D62, D62.1C22, C124.16C22, Dic31.2C22, (C2×C4)⋊3D31, (C2×C124)⋊4C2, (C4×D31)⋊4C2, C31⋊1(C4○D4), C31⋊D4⋊3C2, C2.5(C22×D31), (C2×C62).11C22, SmallGroup(496,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D124⋊5C2
G = < a,b,c | a124=b2=c2=1, bab=a-1, ac=ca, cbc=a62b >
Smallest permutation representation of D124⋊5C2
►On 248 points
Generators in S248
(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)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(125 222)(126 221)(127 220)(128 219)(129 218)(130 217)(131 216)(132 215)(133 214)(134 213)(135 212)(136 211)(137 210)(138 209)(139 208)(140 207)(141 206)(142 205)(143 204)(144 203)(145 202)(146 201)(147 200)(148 199)(149 198)(150 197)(151 196)(152 195)(153 194)(154 193)(155 192)(156 191)(157 190)(158 189)(159 188)(160 187)(161 186)(162 185)(163 184)(164 183)(165 182)(166 181)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(223 248)(224 247)(225 246)(226 245)(227 244)(228 243)(229 242)(230 241)(231 240)(232 239)(233 238)(234 237)(235 236)
(1 205)(2 206)(3 207)(4 208)(5 209)(6 210)(7 211)(8 212)(9 213)(10 214)(11 215)(12 216)(13 217)(14 218)(15 219)(16 220)(17 221)(18 222)(19 223)(20 224)(21 225)(22 226)(23 227)(24 228)(25 229)(26 230)(27 231)(28 232)(29 233)(30 234)(31 235)(32 236)(33 237)(34 238)(35 239)(36 240)(37 241)(38 242)(39 243)(40 244)(41 245)(42 246)(43 247)(44 248)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 161)(82 162)(83 163)(84 164)(85 165)(86 166)(87 167)(88 168)(89 169)(90 170)(91 171)(92 172)(93 173)(94 174)(95 175)(96 176)(97 177)(98 178)(99 179)(100 180)(101 181)(102 182)(103 183)(104 184)(105 185)(106 186)(107 187)(108 188)(109 189)(110 190)(111 191)(112 192)(113 193)(114 194)(115 195)(116 196)(117 197)(118 198)(119 199)(120 200)(121 201)(122 202)(123 203)(124 204)
G:=sub<Sym(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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(223,248)(224,247)(225,246)(226,245)(227,244)(228,243)(229,242)(230,241)(231,240)(232,239)(233,238)(234,237)(235,236), (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,225)(22,226)(23,227)(24,228)(25,229)(26,230)(27,231)(28,232)(29,233)(30,234)(31,235)(32,236)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,161)(82,162)(83,163)(84,164)(85,165)(86,166)(87,167)(88,168)(89,169)(90,170)(91,171)(92,172)(93,173)(94,174)(95,175)(96,176)(97,177)(98,178)(99,179)(100,180)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,193)(114,194)(115,195)(116,196)(117,197)(118,198)(119,199)(120,200)(121,201)(122,202)(123,203)(124,204)>;
G:=Group( (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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(223,248)(224,247)(225,246)(226,245)(227,244)(228,243)(229,242)(230,241)(231,240)(232,239)(233,238)(234,237)(235,236), (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,225)(22,226)(23,227)(24,228)(25,229)(26,230)(27,231)(28,232)(29,233)(30,234)(31,235)(32,236)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,161)(82,162)(83,163)(84,164)(85,165)(86,166)(87,167)(88,168)(89,169)(90,170)(91,171)(92,172)(93,173)(94,174)(95,175)(96,176)(97,177)(98,178)(99,179)(100,180)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,193)(114,194)(115,195)(116,196)(117,197)(118,198)(119,199)(120,200)(121,201)(122,202)(123,203)(124,204) );
G=PermutationGroup([[(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)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,88),(38,87),(39,86),(40,85),(41,84),(42,83),(43,82),(44,81),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(125,222),(126,221),(127,220),(128,219),(129,218),(130,217),(131,216),(132,215),(133,214),(134,213),(135,212),(136,211),(137,210),(138,209),(139,208),(140,207),(141,206),(142,205),(143,204),(144,203),(145,202),(146,201),(147,200),(148,199),(149,198),(150,197),(151,196),(152,195),(153,194),(154,193),(155,192),(156,191),(157,190),(158,189),(159,188),(160,187),(161,186),(162,185),(163,184),(164,183),(165,182),(166,181),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(223,248),(224,247),(225,246),(226,245),(227,244),(228,243),(229,242),(230,241),(231,240),(232,239),(233,238),(234,237),(235,236)], [(1,205),(2,206),(3,207),(4,208),(5,209),(6,210),(7,211),(8,212),(9,213),(10,214),(11,215),(12,216),(13,217),(14,218),(15,219),(16,220),(17,221),(18,222),(19,223),(20,224),(21,225),(22,226),(23,227),(24,228),(25,229),(26,230),(27,231),(28,232),(29,233),(30,234),(31,235),(32,236),(33,237),(34,238),(35,239),(36,240),(37,241),(38,242),(39,243),(40,244),(41,245),(42,246),(43,247),(44,248),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160),(81,161),(82,162),(83,163),(84,164),(85,165),(86,166),(87,167),(88,168),(89,169),(90,170),(91,171),(92,172),(93,173),(94,174),(95,175),(96,176),(97,177),(98,178),(99,179),(100,180),(101,181),(102,182),(103,183),(104,184),(105,185),(106,186),(107,187),(108,188),(109,189),(110,190),(111,191),(112,192),(113,193),(114,194),(115,195),(116,196),(117,197),(118,198),(119,199),(120,200),(121,201),(122,202),(123,203),(124,204)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 62 | 62 | 1 | 1 | 2 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D31 | D62 | D62 | D124⋊5C2 |
| kernel | D124⋊5C2 | Dic62 | C4×D31 | D124 | C31⋊D4 | C2×C124 | C31 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 15 | 30 | 15 | 60 |
Matrix representation of D124⋊5C2 ►in GL2(𝔽373) generated by
| 170 | 74 |
| 267 | 351 |
| 239 | 99 |
| 56 | 134 |
| 202 | 78 |
| 352 | 171 |
G:=sub<GL(2,GF(373))| [170,267,74,351],[239,56,99,134],[202,352,78,171] >;
D124⋊5C2 in GAP, Magma, Sage, TeX
D_{124}\rtimes_5C_2 % in TeX
G:=Group("D124:5C2"); // GroupNames label
G:=SmallGroup(496,30);
// by ID
G=gap.SmallGroup(496,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,46,182,12004]);
// Polycyclic
G:=Group<a,b,c|a^124=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^62*b>;
// generators/relations
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