direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C18×D13, C234⋊2C2, C26⋊3C18, C78.11C6, C117⋊3C22, C3.(C6×D13), C13⋊3(C2×C18), C39.3(C2×C6), C6.3(C3×D13), (C3×D13).8C6, (C6×D13).2C3, SmallGroup(468,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C18×D13 |
Generators and relations for C18×D13
G = < a,b,c | a18=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C18×D13
►On 234 points
Generators in S234
(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 81 28 167 91 185 51 150 124 65 132 203 222)(2 82 29 168 92 186 52 151 125 66 133 204 223)(3 83 30 169 93 187 53 152 126 67 134 205 224)(4 84 31 170 94 188 54 153 109 68 135 206 225)(5 85 32 171 95 189 37 154 110 69 136 207 226)(6 86 33 172 96 190 38 155 111 70 137 208 227)(7 87 34 173 97 191 39 156 112 71 138 209 228)(8 88 35 174 98 192 40 157 113 72 139 210 229)(9 89 36 175 99 193 41 158 114 55 140 211 230)(10 90 19 176 100 194 42 159 115 56 141 212 231)(11 73 20 177 101 195 43 160 116 57 142 213 232)(12 74 21 178 102 196 44 161 117 58 143 214 233)(13 75 22 179 103 197 45 162 118 59 144 215 234)(14 76 23 180 104 198 46 145 119 60 127 216 217)(15 77 24 163 105 181 47 146 120 61 128 199 218)(16 78 25 164 106 182 48 147 121 62 129 200 219)(17 79 26 165 107 183 49 148 122 63 130 201 220)(18 80 27 166 108 184 50 149 123 64 131 202 221)
(1 231)(2 232)(3 233)(4 234)(5 217)(6 218)(7 219)(8 220)(9 221)(10 222)(11 223)(12 224)(13 225)(14 226)(15 227)(16 228)(17 229)(18 230)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 127)(33 128)(34 129)(35 130)(36 131)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 166)(56 167)(57 168)(58 169)(59 170)(60 171)(61 172)(62 173)(63 174)(64 175)(65 176)(66 177)(67 178)(68 179)(69 180)(70 163)(71 164)(72 165)(73 204)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 216)(86 199)(87 200)(88 201)(89 202)(90 203)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 121)(98 122)(99 123)(100 124)(101 125)(102 126)(103 109)(104 110)(105 111)(106 112)(107 113)(108 114)(145 189)(146 190)(147 191)(148 192)(149 193)(150 194)(151 195)(152 196)(153 197)(154 198)(155 181)(156 182)(157 183)(158 184)(159 185)(160 186)(161 187)(162 188)
G:=sub<Sym(234)| (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), (1,81,28,167,91,185,51,150,124,65,132,203,222)(2,82,29,168,92,186,52,151,125,66,133,204,223)(3,83,30,169,93,187,53,152,126,67,134,205,224)(4,84,31,170,94,188,54,153,109,68,135,206,225)(5,85,32,171,95,189,37,154,110,69,136,207,226)(6,86,33,172,96,190,38,155,111,70,137,208,227)(7,87,34,173,97,191,39,156,112,71,138,209,228)(8,88,35,174,98,192,40,157,113,72,139,210,229)(9,89,36,175,99,193,41,158,114,55,140,211,230)(10,90,19,176,100,194,42,159,115,56,141,212,231)(11,73,20,177,101,195,43,160,116,57,142,213,232)(12,74,21,178,102,196,44,161,117,58,143,214,233)(13,75,22,179,103,197,45,162,118,59,144,215,234)(14,76,23,180,104,198,46,145,119,60,127,216,217)(15,77,24,163,105,181,47,146,120,61,128,199,218)(16,78,25,164,106,182,48,147,121,62,129,200,219)(17,79,26,165,107,183,49,148,122,63,130,201,220)(18,80,27,166,108,184,50,149,123,64,131,202,221), (1,231)(2,232)(3,233)(4,234)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,225)(14,226)(15,227)(16,228)(17,229)(18,230)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,127)(33,128)(34,129)(35,130)(36,131)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,163)(71,164)(72,165)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,199)(87,200)(88,201)(89,202)(90,203)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,109)(104,110)(105,111)(106,112)(107,113)(108,114)(145,189)(146,190)(147,191)(148,192)(149,193)(150,194)(151,195)(152,196)(153,197)(154,198)(155,181)(156,182)(157,183)(158,184)(159,185)(160,186)(161,187)(162,188)>;
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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (1,81,28,167,91,185,51,150,124,65,132,203,222)(2,82,29,168,92,186,52,151,125,66,133,204,223)(3,83,30,169,93,187,53,152,126,67,134,205,224)(4,84,31,170,94,188,54,153,109,68,135,206,225)(5,85,32,171,95,189,37,154,110,69,136,207,226)(6,86,33,172,96,190,38,155,111,70,137,208,227)(7,87,34,173,97,191,39,156,112,71,138,209,228)(8,88,35,174,98,192,40,157,113,72,139,210,229)(9,89,36,175,99,193,41,158,114,55,140,211,230)(10,90,19,176,100,194,42,159,115,56,141,212,231)(11,73,20,177,101,195,43,160,116,57,142,213,232)(12,74,21,178,102,196,44,161,117,58,143,214,233)(13,75,22,179,103,197,45,162,118,59,144,215,234)(14,76,23,180,104,198,46,145,119,60,127,216,217)(15,77,24,163,105,181,47,146,120,61,128,199,218)(16,78,25,164,106,182,48,147,121,62,129,200,219)(17,79,26,165,107,183,49,148,122,63,130,201,220)(18,80,27,166,108,184,50,149,123,64,131,202,221), (1,231)(2,232)(3,233)(4,234)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,225)(14,226)(15,227)(16,228)(17,229)(18,230)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,127)(33,128)(34,129)(35,130)(36,131)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,163)(71,164)(72,165)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,199)(87,200)(88,201)(89,202)(90,203)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,109)(104,110)(105,111)(106,112)(107,113)(108,114)(145,189)(146,190)(147,191)(148,192)(149,193)(150,194)(151,195)(152,196)(153,197)(154,198)(155,181)(156,182)(157,183)(158,184)(159,185)(160,186)(161,187)(162,188) );
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),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)], [(1,81,28,167,91,185,51,150,124,65,132,203,222),(2,82,29,168,92,186,52,151,125,66,133,204,223),(3,83,30,169,93,187,53,152,126,67,134,205,224),(4,84,31,170,94,188,54,153,109,68,135,206,225),(5,85,32,171,95,189,37,154,110,69,136,207,226),(6,86,33,172,96,190,38,155,111,70,137,208,227),(7,87,34,173,97,191,39,156,112,71,138,209,228),(8,88,35,174,98,192,40,157,113,72,139,210,229),(9,89,36,175,99,193,41,158,114,55,140,211,230),(10,90,19,176,100,194,42,159,115,56,141,212,231),(11,73,20,177,101,195,43,160,116,57,142,213,232),(12,74,21,178,102,196,44,161,117,58,143,214,233),(13,75,22,179,103,197,45,162,118,59,144,215,234),(14,76,23,180,104,198,46,145,119,60,127,216,217),(15,77,24,163,105,181,47,146,120,61,128,199,218),(16,78,25,164,106,182,48,147,121,62,129,200,219),(17,79,26,165,107,183,49,148,122,63,130,201,220),(18,80,27,166,108,184,50,149,123,64,131,202,221)], [(1,231),(2,232),(3,233),(4,234),(5,217),(6,218),(7,219),(8,220),(9,221),(10,222),(11,223),(12,224),(13,225),(14,226),(15,227),(16,228),(17,229),(18,230),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,127),(33,128),(34,129),(35,130),(36,131),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,166),(56,167),(57,168),(58,169),(59,170),(60,171),(61,172),(62,173),(63,174),(64,175),(65,176),(66,177),(67,178),(68,179),(69,180),(70,163),(71,164),(72,165),(73,204),(74,205),(75,206),(76,207),(77,208),(78,209),(79,210),(80,211),(81,212),(82,213),(83,214),(84,215),(85,216),(86,199),(87,200),(88,201),(89,202),(90,203),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(97,121),(98,122),(99,123),(100,124),(101,125),(102,126),(103,109),(104,110),(105,111),(106,112),(107,113),(108,114),(145,189),(146,190),(147,191),(148,192),(149,193),(150,194),(151,195),(152,196),(153,197),(154,198),(155,181),(156,182),(157,183),(158,184),(159,185),(160,186),(161,187),(162,188)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 13A | ··· | 13F | 18A | ··· | 18F | 18G | ··· | 18R | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 13 | ··· | 13 | 18 | ··· | 18 | 18 | ··· | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D13 | D26 | C3×D13 | C6×D13 | C9×D13 | C18×D13 |
| kernel | C18×D13 | C9×D13 | C234 | C6×D13 | C3×D13 | C78 | D26 | D13 | C26 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 6 | 6 | 12 | 12 | 36 | 36 |
Matrix representation of C18×D13 ►in GL3(𝔽937) generated by
| 924 | 0 | 0 |
| 0 | 323 | 0 |
| 0 | 0 | 323 |
| 1 | 0 | 0 |
| 0 | 904 | 1 |
| 0 | 361 | 131 |
| 936 | 0 | 0 |
| 0 | 806 | 1 |
| 0 | 643 | 131 |
G:=sub<GL(3,GF(937))| [924,0,0,0,323,0,0,0,323],[1,0,0,0,904,361,0,1,131],[936,0,0,0,806,643,0,1,131] >;
C18×D13 in GAP, Magma, Sage, TeX
C_{18}\times D_{13} % in TeX
G:=Group("C18xD13"); // GroupNames label
G:=SmallGroup(468,15);
// by ID
G=gap.SmallGroup(468,15);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,57,10804]);
// Polycyclic
G:=Group<a,b,c|a^18=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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