direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C13⋊C9, C26⋊C9, C78.C3, C13⋊2C18, C39.2C6, C6.(C13⋊C3), C3.(C2×C13⋊C3), SmallGroup(234,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊C9 |
Generators and relations for C2×C13⋊C9
G = < a,b,c | a2=b13=c9=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C2×C13⋊C9
►Regular action on 234 points
Generators in S234
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 138)(22 139)(23 140)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 165)(41 166)(42 167)(43 168)(44 169)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 175)(54 176)(55 177)(56 178)(57 179)(58 180)(59 181)(60 182)(61 170)(62 171)(63 172)(64 173)(65 174)(66 195)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 191)(76 192)(77 193)(78 194)(79 208)(80 196)(81 197)(82 198)(83 199)(84 200)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(101 218)(102 219)(103 220)(104 221)(105 234)(106 222)(107 223)(108 224)(109 225)(110 226)(111 227)(112 228)(113 229)(114 230)(115 231)(116 232)(117 233)
(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)
(1 113 69 27 94 60 14 85 44)(2 116 78 28 97 56 15 88 40)(3 106 74 29 100 65 16 91 49)(4 109 70 30 103 61 17 81 45)(5 112 66 31 93 57 18 84 41)(6 115 75 32 96 53 19 87 50)(7 105 71 33 99 62 20 90 46)(8 108 67 34 102 58 21 80 42)(9 111 76 35 92 54 22 83 51)(10 114 72 36 95 63 23 86 47)(11 117 68 37 98 59 24 89 43)(12 107 77 38 101 55 25 79 52)(13 110 73 39 104 64 26 82 48)(118 229 185 144 211 182 131 201 169)(119 232 194 145 214 178 132 204 165)(120 222 190 146 217 174 133 207 161)(121 225 186 147 220 170 134 197 157)(122 228 195 148 210 179 135 200 166)(123 231 191 149 213 175 136 203 162)(124 234 187 150 216 171 137 206 158)(125 224 183 151 219 180 138 196 167)(126 227 192 152 209 176 139 199 163)(127 230 188 153 212 172 140 202 159)(128 233 184 154 215 181 141 205 168)(129 223 193 155 218 177 142 208 164)(130 226 189 156 221 173 143 198 160)
G:=sub<Sym(234)| (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,165)(41,166)(42,167)(43,168)(44,169)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,181)(60,182)(61,170)(62,171)(63,172)(64,173)(65,174)(66,195)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,208)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,234)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233), (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), (1,113,69,27,94,60,14,85,44)(2,116,78,28,97,56,15,88,40)(3,106,74,29,100,65,16,91,49)(4,109,70,30,103,61,17,81,45)(5,112,66,31,93,57,18,84,41)(6,115,75,32,96,53,19,87,50)(7,105,71,33,99,62,20,90,46)(8,108,67,34,102,58,21,80,42)(9,111,76,35,92,54,22,83,51)(10,114,72,36,95,63,23,86,47)(11,117,68,37,98,59,24,89,43)(12,107,77,38,101,55,25,79,52)(13,110,73,39,104,64,26,82,48)(118,229,185,144,211,182,131,201,169)(119,232,194,145,214,178,132,204,165)(120,222,190,146,217,174,133,207,161)(121,225,186,147,220,170,134,197,157)(122,228,195,148,210,179,135,200,166)(123,231,191,149,213,175,136,203,162)(124,234,187,150,216,171,137,206,158)(125,224,183,151,219,180,138,196,167)(126,227,192,152,209,176,139,199,163)(127,230,188,153,212,172,140,202,159)(128,233,184,154,215,181,141,205,168)(129,223,193,155,218,177,142,208,164)(130,226,189,156,221,173,143,198,160)>;
G:=Group( (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,165)(41,166)(42,167)(43,168)(44,169)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,181)(60,182)(61,170)(62,171)(63,172)(64,173)(65,174)(66,195)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,208)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,234)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,229)(114,230)(115,231)(116,232)(117,233), (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), (1,113,69,27,94,60,14,85,44)(2,116,78,28,97,56,15,88,40)(3,106,74,29,100,65,16,91,49)(4,109,70,30,103,61,17,81,45)(5,112,66,31,93,57,18,84,41)(6,115,75,32,96,53,19,87,50)(7,105,71,33,99,62,20,90,46)(8,108,67,34,102,58,21,80,42)(9,111,76,35,92,54,22,83,51)(10,114,72,36,95,63,23,86,47)(11,117,68,37,98,59,24,89,43)(12,107,77,38,101,55,25,79,52)(13,110,73,39,104,64,26,82,48)(118,229,185,144,211,182,131,201,169)(119,232,194,145,214,178,132,204,165)(120,222,190,146,217,174,133,207,161)(121,225,186,147,220,170,134,197,157)(122,228,195,148,210,179,135,200,166)(123,231,191,149,213,175,136,203,162)(124,234,187,150,216,171,137,206,158)(125,224,183,151,219,180,138,196,167)(126,227,192,152,209,176,139,199,163)(127,230,188,153,212,172,140,202,159)(128,233,184,154,215,181,141,205,168)(129,223,193,155,218,177,142,208,164)(130,226,189,156,221,173,143,198,160) );
G=PermutationGroup([[(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,138),(22,139),(23,140),(24,141),(25,142),(26,143),(27,144),(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,165),(41,166),(42,167),(43,168),(44,169),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,175),(54,176),(55,177),(56,178),(57,179),(58,180),(59,181),(60,182),(61,170),(62,171),(63,172),(64,173),(65,174),(66,195),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(73,189),(74,190),(75,191),(76,192),(77,193),(78,194),(79,208),(80,196),(81,197),(82,198),(83,199),(84,200),(85,201),(86,202),(87,203),(88,204),(89,205),(90,206),(91,207),(92,209),(93,210),(94,211),(95,212),(96,213),(97,214),(98,215),(99,216),(100,217),(101,218),(102,219),(103,220),(104,221),(105,234),(106,222),(107,223),(108,224),(109,225),(110,226),(111,227),(112,228),(113,229),(114,230),(115,231),(116,232),(117,233)], [(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)], [(1,113,69,27,94,60,14,85,44),(2,116,78,28,97,56,15,88,40),(3,106,74,29,100,65,16,91,49),(4,109,70,30,103,61,17,81,45),(5,112,66,31,93,57,18,84,41),(6,115,75,32,96,53,19,87,50),(7,105,71,33,99,62,20,90,46),(8,108,67,34,102,58,21,80,42),(9,111,76,35,92,54,22,83,51),(10,114,72,36,95,63,23,86,47),(11,117,68,37,98,59,24,89,43),(12,107,77,38,101,55,25,79,52),(13,110,73,39,104,64,26,82,48),(118,229,185,144,211,182,131,201,169),(119,232,194,145,214,178,132,204,165),(120,222,190,146,217,174,133,207,161),(121,225,186,147,220,170,134,197,157),(122,228,195,148,210,179,135,200,166),(123,231,191,149,213,175,136,203,162),(124,234,187,150,216,171,137,206,158),(125,224,183,151,219,180,138,196,167),(126,227,192,152,209,176,139,199,163),(127,230,188,153,212,172,140,202,159),(128,233,184,154,215,181,141,205,168),(129,223,193,155,218,177,142,208,164),(130,226,189,156,221,173,143,198,160)]])
C2×C13⋊C9 is a maximal subgroup of
C13⋊2C36
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 13A | 13B | 13C | 13D | 18A | ··· | 18F | 26A | 26B | 26C | 26D | 39A | ··· | 39H | 78A | ··· | 78H |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 13 | 13 | 13 | 13 | 18 | ··· | 18 | 26 | 26 | 26 | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C13⋊C3 | C2×C13⋊C3 | C13⋊C9 | C2×C13⋊C9 |
| kernel | C2×C13⋊C9 | C13⋊C9 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 4 | 4 | 8 | 8 |
Matrix representation of C2×C13⋊C9 ►in GL4(𝔽937) generated by
| 936 | 0 | 0 | 0 |
| 0 | 936 | 0 | 0 |
| 0 | 0 | 936 | 0 |
| 0 | 0 | 0 | 936 |
| 1 | 0 | 0 | 0 |
| 0 | 661 | 482 | 1 |
| 0 | 662 | 482 | 1 |
| 0 | 661 | 483 | 1 |
| 169 | 0 | 0 | 0 |
| 0 | 857 | 171 | 187 |
| 0 | 588 | 771 | 646 |
| 0 | 887 | 392 | 246 |
G:=sub<GL(4,GF(937))| [936,0,0,0,0,936,0,0,0,0,936,0,0,0,0,936],[1,0,0,0,0,661,662,661,0,482,482,483,0,1,1,1],[169,0,0,0,0,857,588,887,0,171,771,392,0,187,646,246] >;
C2×C13⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes C_9 % in TeX
G:=Group("C2xC13:C9"); // GroupNames label
G:=SmallGroup(234,2);
// by ID
G=gap.SmallGroup(234,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,29,439]);
// Polycyclic
G:=Group<a,b,c|a^2=b^13=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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