direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C46, C10⋊C46, C230⋊3C2, C115⋊4C22, C5⋊(C2×C46), SmallGroup(460,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C46 |
Generators and relations for D5×C46
G = < a,b,c | a46=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C46
►On 230 points
Generators in S230
(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)
(1 180 103 208 67)(2 181 104 209 68)(3 182 105 210 69)(4 183 106 211 70)(5 184 107 212 71)(6 139 108 213 72)(7 140 109 214 73)(8 141 110 215 74)(9 142 111 216 75)(10 143 112 217 76)(11 144 113 218 77)(12 145 114 219 78)(13 146 115 220 79)(14 147 116 221 80)(15 148 117 222 81)(16 149 118 223 82)(17 150 119 224 83)(18 151 120 225 84)(19 152 121 226 85)(20 153 122 227 86)(21 154 123 228 87)(22 155 124 229 88)(23 156 125 230 89)(24 157 126 185 90)(25 158 127 186 91)(26 159 128 187 92)(27 160 129 188 47)(28 161 130 189 48)(29 162 131 190 49)(30 163 132 191 50)(31 164 133 192 51)(32 165 134 193 52)(33 166 135 194 53)(34 167 136 195 54)(35 168 137 196 55)(36 169 138 197 56)(37 170 93 198 57)(38 171 94 199 58)(39 172 95 200 59)(40 173 96 201 60)(41 174 97 202 61)(42 175 98 203 62)(43 176 99 204 63)(44 177 100 205 64)(45 178 101 206 65)(46 179 102 207 66)
(1 90)(2 91)(3 92)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)(169 220)(170 221)(171 222)(172 223)(173 224)(174 225)(175 226)(176 227)(177 228)(178 229)(179 230)(180 185)(181 186)(182 187)(183 188)(184 189)
G:=sub<Sym(230)| (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), (1,180,103,208,67)(2,181,104,209,68)(3,182,105,210,69)(4,183,106,211,70)(5,184,107,212,71)(6,139,108,213,72)(7,140,109,214,73)(8,141,110,215,74)(9,142,111,216,75)(10,143,112,217,76)(11,144,113,218,77)(12,145,114,219,78)(13,146,115,220,79)(14,147,116,221,80)(15,148,117,222,81)(16,149,118,223,82)(17,150,119,224,83)(18,151,120,225,84)(19,152,121,226,85)(20,153,122,227,86)(21,154,123,228,87)(22,155,124,229,88)(23,156,125,230,89)(24,157,126,185,90)(25,158,127,186,91)(26,159,128,187,92)(27,160,129,188,47)(28,161,130,189,48)(29,162,131,190,49)(30,163,132,191,50)(31,164,133,192,51)(32,165,134,193,52)(33,166,135,194,53)(34,167,136,195,54)(35,168,137,196,55)(36,169,138,197,56)(37,170,93,198,57)(38,171,94,199,58)(39,172,95,200,59)(40,173,96,201,60)(41,174,97,202,61)(42,175,98,203,62)(43,176,99,204,63)(44,177,100,205,64)(45,178,101,206,65)(46,179,102,207,66), (1,90)(2,91)(3,92)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,185)(181,186)(182,187)(183,188)(184,189)>;
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), (1,180,103,208,67)(2,181,104,209,68)(3,182,105,210,69)(4,183,106,211,70)(5,184,107,212,71)(6,139,108,213,72)(7,140,109,214,73)(8,141,110,215,74)(9,142,111,216,75)(10,143,112,217,76)(11,144,113,218,77)(12,145,114,219,78)(13,146,115,220,79)(14,147,116,221,80)(15,148,117,222,81)(16,149,118,223,82)(17,150,119,224,83)(18,151,120,225,84)(19,152,121,226,85)(20,153,122,227,86)(21,154,123,228,87)(22,155,124,229,88)(23,156,125,230,89)(24,157,126,185,90)(25,158,127,186,91)(26,159,128,187,92)(27,160,129,188,47)(28,161,130,189,48)(29,162,131,190,49)(30,163,132,191,50)(31,164,133,192,51)(32,165,134,193,52)(33,166,135,194,53)(34,167,136,195,54)(35,168,137,196,55)(36,169,138,197,56)(37,170,93,198,57)(38,171,94,199,58)(39,172,95,200,59)(40,173,96,201,60)(41,174,97,202,61)(42,175,98,203,62)(43,176,99,204,63)(44,177,100,205,64)(45,178,101,206,65)(46,179,102,207,66), (1,90)(2,91)(3,92)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,185)(181,186)(182,187)(183,188)(184,189) );
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)], [(1,180,103,208,67),(2,181,104,209,68),(3,182,105,210,69),(4,183,106,211,70),(5,184,107,212,71),(6,139,108,213,72),(7,140,109,214,73),(8,141,110,215,74),(9,142,111,216,75),(10,143,112,217,76),(11,144,113,218,77),(12,145,114,219,78),(13,146,115,220,79),(14,147,116,221,80),(15,148,117,222,81),(16,149,118,223,82),(17,150,119,224,83),(18,151,120,225,84),(19,152,121,226,85),(20,153,122,227,86),(21,154,123,228,87),(22,155,124,229,88),(23,156,125,230,89),(24,157,126,185,90),(25,158,127,186,91),(26,159,128,187,92),(27,160,129,188,47),(28,161,130,189,48),(29,162,131,190,49),(30,163,132,191,50),(31,164,133,192,51),(32,165,134,193,52),(33,166,135,194,53),(34,167,136,195,54),(35,168,137,196,55),(36,169,138,197,56),(37,170,93,198,57),(38,171,94,199,58),(39,172,95,200,59),(40,173,96,201,60),(41,174,97,202,61),(42,175,98,203,62),(43,176,99,204,63),(44,177,100,205,64),(45,178,101,206,65),(46,179,102,207,66)], [(1,90),(2,91),(3,92),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(139,190),(140,191),(141,192),(142,193),(143,194),(144,195),(145,196),(146,197),(147,198),(148,199),(149,200),(150,201),(151,202),(152,203),(153,204),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213),(163,214),(164,215),(165,216),(166,217),(167,218),(168,219),(169,220),(170,221),(171,222),(172,223),(173,224),(174,225),(175,226),(176,227),(177,228),(178,229),(179,230),(180,185),(181,186),(182,187),(183,188),(184,189)]])
184 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46BN | 115A | ··· | 115AR | 230A | ··· | 230AR |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 115 | ··· | 115 | 230 | ··· | 230 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
184 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C23 | C46 | C46 | D5 | D10 | D5×C23 | D5×C46 |
| kernel | D5×C46 | D5×C23 | C230 | D10 | D5 | C10 | C46 | C23 | C2 | C1 |
| # reps | 1 | 2 | 1 | 22 | 44 | 22 | 2 | 2 | 44 | 44 |
Matrix representation of D5×C46 ►in GL2(𝔽461) generated by
| 139 | 0 |
| 0 | 139 |
| 0 | 1 |
| 460 | 439 |
| 460 | 0 |
| 22 | 1 |
G:=sub<GL(2,GF(461))| [139,0,0,139],[0,460,1,439],[460,22,0,1] >;
D5×C46 in GAP, Magma, Sage, TeX
D_5\times C_{46} % in TeX
G:=Group("D5xC46"); // GroupNames label
G:=SmallGroup(460,9);
// by ID
G=gap.SmallGroup(460,9);
# by ID
G:=PCGroup([4,-2,-2,-23,-5,5891]);
// Polycyclic
G:=Group<a,b,c|a^46=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export