direct product, abelian, monomial, 2-elementary
Aliases: C22×C58, SmallGroup(232,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C58 |
| C1 — C22×C58 |
| C1 — C22×C58 |
Generators and relations for C22×C58
G = < a,b,c | a2=b2=c58=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C58
►Regular action on 232 points
Generators in S232
(1 174)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(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)(59 226)(60 227)(61 228)(62 229)(63 230)(64 231)(65 232)(66 175)(67 176)(68 177)(69 178)(70 179)(71 180)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 196)(88 197)(89 198)(90 199)(91 200)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 209)(101 210)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 220)(112 221)(113 222)(114 223)(115 224)(116 225)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(117 208)(118 209)(119 210)(120 211)(121 212)(122 213)(123 214)(124 215)(125 216)(126 217)(127 218)(128 219)(129 220)(130 221)(131 222)(132 223)(133 224)(134 225)(135 226)(136 227)(137 228)(138 229)(139 230)(140 231)(141 232)(142 175)(143 176)(144 177)(145 178)(146 179)(147 180)(148 181)(149 182)(150 183)(151 184)(152 185)(153 186)(154 187)(155 188)(156 189)(157 190)(158 191)(159 192)(160 193)(161 194)(162 195)(163 196)(164 197)(165 198)(166 199)(167 200)(168 201)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)
(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)
G:=sub<Sym(232)| (1,174)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(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)(59,226)(60,227)(61,228)(62,229)(63,230)(64,231)(65,232)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,209)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,220)(112,221)(113,222)(114,223)(115,224)(116,225), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(117,208)(118,209)(119,210)(120,211)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,225)(135,226)(136,227)(137,228)(138,229)(139,230)(140,231)(141,232)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207), (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)>;
G:=Group( (1,174)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(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)(59,226)(60,227)(61,228)(62,229)(63,230)(64,231)(65,232)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,209)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,220)(112,221)(113,222)(114,223)(115,224)(116,225), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(117,208)(118,209)(119,210)(120,211)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,225)(135,226)(136,227)(137,228)(138,229)(139,230)(140,231)(141,232)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207), (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) );
G=PermutationGroup([[(1,174),(2,117),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(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),(59,226),(60,227),(61,228),(62,229),(63,230),(64,231),(65,232),(66,175),(67,176),(68,177),(69,178),(70,179),(71,180),(72,181),(73,182),(74,183),(75,184),(76,185),(77,186),(78,187),(79,188),(80,189),(81,190),(82,191),(83,192),(84,193),(85,194),(86,195),(87,196),(88,197),(89,198),(90,199),(91,200),(92,201),(93,202),(94,203),(95,204),(96,205),(97,206),(98,207),(99,208),(100,209),(101,210),(102,211),(103,212),(104,213),(105,214),(106,215),(107,216),(108,217),(109,218),(110,219),(111,220),(112,221),(113,222),(114,223),(115,224),(116,225)], [(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(117,208),(118,209),(119,210),(120,211),(121,212),(122,213),(123,214),(124,215),(125,216),(126,217),(127,218),(128,219),(129,220),(130,221),(131,222),(132,223),(133,224),(134,225),(135,226),(136,227),(137,228),(138,229),(139,230),(140,231),(141,232),(142,175),(143,176),(144,177),(145,178),(146,179),(147,180),(148,181),(149,182),(150,183),(151,184),(152,185),(153,186),(154,187),(155,188),(156,189),(157,190),(158,191),(159,192),(160,193),(161,194),(162,195),(163,196),(164,197),(165,198),(166,199),(167,200),(168,201),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207)], [(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)]])
C22×C58 is a maximal subgroup of
C23.D29
232 conjugacy classes
| class | 1 | 2A | ··· | 2G | 29A | ··· | 29AB | 58A | ··· | 58GN |
| order | 1 | 2 | ··· | 2 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
232 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C29 | C58 |
| kernel | C22×C58 | C2×C58 | C23 | C22 |
| # reps | 1 | 7 | 28 | 196 |
Matrix representation of C22×C58 ►in GL3(𝔽59) generated by
| 58 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 58 |
| 1 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 1 |
| 44 | 0 | 0 |
| 0 | 43 | 0 |
| 0 | 0 | 35 |
G:=sub<GL(3,GF(59))| [58,0,0,0,1,0,0,0,58],[1,0,0,0,58,0,0,0,1],[44,0,0,0,43,0,0,0,35] >;
C22×C58 in GAP, Magma, Sage, TeX
C_2^2\times C_{58} % in TeX
G:=Group("C2^2xC58"); // GroupNames label
G:=SmallGroup(232,14);
// by ID
G=gap.SmallGroup(232,14);
# by ID
G:=PCGroup([4,-2,-2,-2,-29]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^58=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export