direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D23, C46⋊C10, C230⋊2C2, C115⋊3C22, C23⋊(C2×C10), SmallGroup(460,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C10×D23 |
Generators and relations for C10×D23
G = < a,b,c | a10=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D23
►On 230 points
Generators in S230
(1 169 98 147 71 134 67 223 37 205)(2 170 99 148 72 135 68 224 38 206)(3 171 100 149 73 136 69 225 39 207)(4 172 101 150 74 137 47 226 40 185)(5 173 102 151 75 138 48 227 41 186)(6 174 103 152 76 116 49 228 42 187)(7 175 104 153 77 117 50 229 43 188)(8 176 105 154 78 118 51 230 44 189)(9 177 106 155 79 119 52 208 45 190)(10 178 107 156 80 120 53 209 46 191)(11 179 108 157 81 121 54 210 24 192)(12 180 109 158 82 122 55 211 25 193)(13 181 110 159 83 123 56 212 26 194)(14 182 111 160 84 124 57 213 27 195)(15 183 112 161 85 125 58 214 28 196)(16 184 113 139 86 126 59 215 29 197)(17 162 114 140 87 127 60 216 30 198)(18 163 115 141 88 128 61 217 31 199)(19 164 93 142 89 129 62 218 32 200)(20 165 94 143 90 130 63 219 33 201)(21 166 95 144 91 131 64 220 34 202)(22 167 96 145 92 132 65 221 35 203)(23 168 97 146 70 133 66 222 36 204)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 120)(15 119)(16 118)(17 117)(18 116)(19 138)(20 137)(21 136)(22 135)(23 134)(24 159)(25 158)(26 157)(27 156)(28 155)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(41 142)(42 141)(43 140)(44 139)(45 161)(46 160)(47 165)(48 164)(49 163)(50 162)(51 184)(52 183)(53 182)(54 181)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 168)(68 167)(69 166)(70 205)(71 204)(72 203)(73 202)(74 201)(75 200)(76 199)(77 198)(78 197)(79 196)(80 195)(81 194)(82 193)(83 192)(84 191)(85 190)(86 189)(87 188)(88 187)(89 186)(90 185)(91 207)(92 206)(93 227)(94 226)(95 225)(96 224)(97 223)(98 222)(99 221)(100 220)(101 219)(102 218)(103 217)(104 216)(105 215)(106 214)(107 213)(108 212)(109 211)(110 210)(111 209)(112 208)(113 230)(114 229)(115 228)
G:=sub<Sym(230)| (1,169,98,147,71,134,67,223,37,205)(2,170,99,148,72,135,68,224,38,206)(3,171,100,149,73,136,69,225,39,207)(4,172,101,150,74,137,47,226,40,185)(5,173,102,151,75,138,48,227,41,186)(6,174,103,152,76,116,49,228,42,187)(7,175,104,153,77,117,50,229,43,188)(8,176,105,154,78,118,51,230,44,189)(9,177,106,155,79,119,52,208,45,190)(10,178,107,156,80,120,53,209,46,191)(11,179,108,157,81,121,54,210,24,192)(12,180,109,158,82,122,55,211,25,193)(13,181,110,159,83,123,56,212,26,194)(14,182,111,160,84,124,57,213,27,195)(15,183,112,161,85,125,58,214,28,196)(16,184,113,139,86,126,59,215,29,197)(17,162,114,140,87,127,60,216,30,198)(18,163,115,141,88,128,61,217,31,199)(19,164,93,142,89,129,62,218,32,200)(20,165,94,143,90,130,63,219,33,201)(21,166,95,144,91,131,64,220,34,202)(22,167,96,145,92,132,65,221,35,203)(23,168,97,146,70,133,66,222,36,204), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,119)(16,118)(17,117)(18,116)(19,138)(20,137)(21,136)(22,135)(23,134)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,161)(46,160)(47,165)(48,164)(49,163)(50,162)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,205)(71,204)(72,203)(73,202)(74,201)(75,200)(76,199)(77,198)(78,197)(79,196)(80,195)(81,194)(82,193)(83,192)(84,191)(85,190)(86,189)(87,188)(88,187)(89,186)(90,185)(91,207)(92,206)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208)(113,230)(114,229)(115,228)>;
G:=Group( (1,169,98,147,71,134,67,223,37,205)(2,170,99,148,72,135,68,224,38,206)(3,171,100,149,73,136,69,225,39,207)(4,172,101,150,74,137,47,226,40,185)(5,173,102,151,75,138,48,227,41,186)(6,174,103,152,76,116,49,228,42,187)(7,175,104,153,77,117,50,229,43,188)(8,176,105,154,78,118,51,230,44,189)(9,177,106,155,79,119,52,208,45,190)(10,178,107,156,80,120,53,209,46,191)(11,179,108,157,81,121,54,210,24,192)(12,180,109,158,82,122,55,211,25,193)(13,181,110,159,83,123,56,212,26,194)(14,182,111,160,84,124,57,213,27,195)(15,183,112,161,85,125,58,214,28,196)(16,184,113,139,86,126,59,215,29,197)(17,162,114,140,87,127,60,216,30,198)(18,163,115,141,88,128,61,217,31,199)(19,164,93,142,89,129,62,218,32,200)(20,165,94,143,90,130,63,219,33,201)(21,166,95,144,91,131,64,220,34,202)(22,167,96,145,92,132,65,221,35,203)(23,168,97,146,70,133,66,222,36,204), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,119)(16,118)(17,117)(18,116)(19,138)(20,137)(21,136)(22,135)(23,134)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,161)(46,160)(47,165)(48,164)(49,163)(50,162)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,205)(71,204)(72,203)(73,202)(74,201)(75,200)(76,199)(77,198)(78,197)(79,196)(80,195)(81,194)(82,193)(83,192)(84,191)(85,190)(86,189)(87,188)(88,187)(89,186)(90,185)(91,207)(92,206)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208)(113,230)(114,229)(115,228) );
G=PermutationGroup([[(1,169,98,147,71,134,67,223,37,205),(2,170,99,148,72,135,68,224,38,206),(3,171,100,149,73,136,69,225,39,207),(4,172,101,150,74,137,47,226,40,185),(5,173,102,151,75,138,48,227,41,186),(6,174,103,152,76,116,49,228,42,187),(7,175,104,153,77,117,50,229,43,188),(8,176,105,154,78,118,51,230,44,189),(9,177,106,155,79,119,52,208,45,190),(10,178,107,156,80,120,53,209,46,191),(11,179,108,157,81,121,54,210,24,192),(12,180,109,158,82,122,55,211,25,193),(13,181,110,159,83,123,56,212,26,194),(14,182,111,160,84,124,57,213,27,195),(15,183,112,161,85,125,58,214,28,196),(16,184,113,139,86,126,59,215,29,197),(17,162,114,140,87,127,60,216,30,198),(18,163,115,141,88,128,61,217,31,199),(19,164,93,142,89,129,62,218,32,200),(20,165,94,143,90,130,63,219,33,201),(21,166,95,144,91,131,64,220,34,202),(22,167,96,145,92,132,65,221,35,203),(23,168,97,146,70,133,66,222,36,204)], [(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)], [(1,133),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,120),(15,119),(16,118),(17,117),(18,116),(19,138),(20,137),(21,136),(22,135),(23,134),(24,159),(25,158),(26,157),(27,156),(28,155),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,143),(41,142),(42,141),(43,140),(44,139),(45,161),(46,160),(47,165),(48,164),(49,163),(50,162),(51,184),(52,183),(53,182),(54,181),(55,180),(56,179),(57,178),(58,177),(59,176),(60,175),(61,174),(62,173),(63,172),(64,171),(65,170),(66,169),(67,168),(68,167),(69,166),(70,205),(71,204),(72,203),(73,202),(74,201),(75,200),(76,199),(77,198),(78,197),(79,196),(80,195),(81,194),(82,193),(83,192),(84,191),(85,190),(86,189),(87,188),(88,187),(89,186),(90,185),(91,207),(92,206),(93,227),(94,226),(95,225),(96,224),(97,223),(98,222),(99,221),(100,220),(101,219),(102,218),(103,217),(104,216),(105,215),(106,214),(107,213),(108,212),(109,211),(110,210),(111,209),(112,208),(113,230),(114,229),(115,228)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 23A | ··· | 23K | 46A | ··· | 46K | 115A | ··· | 115AR | 230A | ··· | 230AR |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 23 | ··· | 23 | 46 | ··· | 46 | 115 | ··· | 115 | 230 | ··· | 230 |
| size | 1 | 1 | 23 | 23 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 23 | ··· | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D23 | D46 | C5×D23 | C10×D23 |
| kernel | C10×D23 | C5×D23 | C230 | D46 | D23 | C46 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 11 | 11 | 44 | 44 |
Matrix representation of C10×D23 ►in GL3(𝔽461) generated by
| 347 | 0 | 0 |
| 0 | 460 | 0 |
| 0 | 0 | 460 |
| 1 | 0 | 0 |
| 0 | 454 | 1 |
| 0 | 175 | 370 |
| 1 | 0 | 0 |
| 0 | 91 | 1 |
| 0 | 18 | 370 |
G:=sub<GL(3,GF(461))| [347,0,0,0,460,0,0,0,460],[1,0,0,0,454,175,0,1,370],[1,0,0,0,91,18,0,1,370] >;
C10×D23 in GAP, Magma, Sage, TeX
C_{10}\times D_{23} % in TeX
G:=Group("C10xD23"); // GroupNames label
G:=SmallGroup(460,8);
// by ID
G=gap.SmallGroup(460,8);
# by ID
G:=PCGroup([4,-2,-2,-5,-23,7043]);
// Polycyclic
G:=Group<a,b,c|a^10=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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