direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D116, C4⋊2D58, C58⋊1D4, C116⋊2C22, D58⋊1C22, C58.3C23, C22.10D58, C29⋊1(C2×D4), (C2×C4)⋊2D29, (C2×C116)⋊3C2, (C22×D29)⋊1C2, C2.4(C22×D29), (C2×C58).10C22, SmallGroup(464,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D116
G = < a,b,c | a2=b116=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 826 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C2×D4, C29, D29, C58, C58, C116, D58, D58, C2×C58, D116, C2×C116, C22×D29, C2×D116
Quotients: C1, C2, C22, D4, C23, C2×D4, D29, D58, D116, C22×D29, C2×D116
►On 232 points
Generators in S232
(1 212)(2 213)(3 214)(4 215)(5 216)(6 217)(7 218)(8 219)(9 220)(10 221)(11 222)(12 223)(13 224)(14 225)(15 226)(16 227)(17 228)(18 229)(19 230)(20 231)(21 232)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 157)(63 158)(64 159)(65 160)(66 161)(67 162)(68 163)(69 164)(70 165)(71 166)(72 167)(73 168)(74 169)(75 170)(76 171)(77 172)(78 173)(79 174)(80 175)(81 176)(82 177)(83 178)(84 179)(85 180)(86 181)(87 182)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 192)(98 193)(99 194)(100 195)(101 196)(102 197)(103 198)(104 199)(105 200)(106 201)(107 202)(108 203)(109 204)(110 205)(111 206)(112 207)(113 208)(114 209)(115 210)(116 211)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(117 190)(118 189)(119 188)(120 187)(121 186)(122 185)(123 184)(124 183)(125 182)(126 181)(127 180)(128 179)(129 178)(130 177)(131 176)(132 175)(133 174)(134 173)(135 172)(136 171)(137 170)(138 169)(139 168)(140 167)(141 166)(142 165)(143 164)(144 163)(145 162)(146 161)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(191 232)(192 231)(193 230)(194 229)(195 228)(196 227)(197 226)(198 225)(199 224)(200 223)(201 222)(202 221)(203 220)(204 219)(205 218)(206 217)(207 216)(208 215)(209 214)(210 213)(211 212)
G:=sub<Sym(232)| (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,157)(63,158)(64,159)(65,160)(66,161)(67,162)(68,163)(69,164)(70,165)(71,166)(72,167)(73,168)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,180)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,192)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,208)(114,209)(115,210)(116,211), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213)(211,212)>;
G:=Group( (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,157)(63,158)(64,159)(65,160)(66,161)(67,162)(68,163)(69,164)(70,165)(71,166)(72,167)(73,168)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,180)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,192)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,208)(114,209)(115,210)(116,211), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213)(211,212) );
G=PermutationGroup([[(1,212),(2,213),(3,214),(4,215),(5,216),(6,217),(7,218),(8,219),(9,220),(10,221),(11,222),(12,223),(13,224),(14,225),(15,226),(16,227),(17,228),(18,229),(19,230),(20,231),(21,232),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,127),(33,128),(34,129),(35,130),(36,131),(37,132),(38,133),(39,134),(40,135),(41,136),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,144),(50,145),(51,146),(52,147),(53,148),(54,149),(55,150),(56,151),(57,152),(58,153),(59,154),(60,155),(61,156),(62,157),(63,158),(64,159),(65,160),(66,161),(67,162),(68,163),(69,164),(70,165),(71,166),(72,167),(73,168),(74,169),(75,170),(76,171),(77,172),(78,173),(79,174),(80,175),(81,176),(82,177),(83,178),(84,179),(85,180),(86,181),(87,182),(88,183),(89,184),(90,185),(91,186),(92,187),(93,188),(94,189),(95,190),(96,191),(97,192),(98,193),(99,194),(100,195),(101,196),(102,197),(103,198),(104,199),(105,200),(106,201),(107,202),(108,203),(109,204),(110,205),(111,206),(112,207),(113,208),(114,209),(115,210),(116,211)], [(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)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,84),(34,83),(35,82),(36,81),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(117,190),(118,189),(119,188),(120,187),(121,186),(122,185),(123,184),(124,183),(125,182),(126,181),(127,180),(128,179),(129,178),(130,177),(131,176),(132,175),(133,174),(134,173),(135,172),(136,171),(137,170),(138,169),(139,168),(140,167),(141,166),(142,165),(143,164),(144,163),(145,162),(146,161),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154),(191,232),(192,231),(193,230),(194,229),(195,228),(196,227),(197,226),(198,225),(199,224),(200,223),(201,222),(202,221),(203,220),(204,219),(205,218),(206,217),(207,216),(208,215),(209,214),(210,213),(211,212)]])
122 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 29A | ··· | 29N | 58A | ··· | 58AP | 116A | ··· | 116BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 58 | 58 | 58 | 58 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
122 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D29 | D58 | D58 | D116 |
| kernel | C2×D116 | D116 | C2×C116 | C22×D29 | C58 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 14 | 28 | 14 | 56 |
Matrix representation of C2×D116 ►in GL3(𝔽233) generated by
| 232 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 224 | 138 |
| 0 | 95 | 226 |
| 232 | 0 | 0 |
| 0 | 224 | 138 |
| 0 | 148 | 9 |
G:=sub<GL(3,GF(233))| [232,0,0,0,1,0,0,0,1],[1,0,0,0,224,95,0,138,226],[232,0,0,0,224,148,0,138,9] >;
C2×D116 in GAP, Magma, Sage, TeX
C_2\times D_{116} % in TeX
G:=Group("C2xD116"); // GroupNames label
G:=SmallGroup(464,37);
// by ID
G=gap.SmallGroup(464,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,182,42,11204]);
// Polycyclic
G:=Group<a,b,c|a^2=b^116=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations