direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C13⋊C18, D26⋊C9, C26⋊C18, D13⋊C18, C78.3C6, C13⋊(C2×C18), C13⋊C9⋊C22, C39.(C2×C6), (C6×D13).C3, C6.3(C13⋊C6), (C3×D13).4C6, (C2×C13⋊C9)⋊C2, C3.(C2×C13⋊C6), SmallGroup(468,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊C18 |
Generators and relations for C2×C13⋊C18
G = < a,b,c | a2=b13=c18=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C2×C13⋊C18
►On 234 points
Generators in S234
(1 18)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 212)(38 213)(39 214)(40 215)(41 216)(42 199)(43 200)(44 201)(45 202)(46 203)(47 204)(48 205)(49 206)(50 207)(51 208)(52 209)(53 210)(54 211)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 109)(71 110)(72 111)(91 168)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(99 176)(100 177)(101 178)(102 179)(103 180)(104 163)(105 164)(106 165)(107 166)(108 167)(127 194)(128 195)(129 196)(130 197)(131 198)(132 181)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 193)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 225)(153 226)(154 227)(155 228)(156 229)(157 230)(158 231)(159 232)(160 233)(161 234)(162 217)
(1 56 163 140 89 50 219 228 41 80 131 172 65)(2 132 229 90 57 173 42 51 164 66 81 220 141)(3 82 52 58 133 221 165 174 230 142 67 43 73)(4 68 175 134 83 44 231 222 53 74 143 166 59)(5 144 223 84 69 167 54 45 176 60 75 232 135)(6 76 46 70 127 233 177 168 224 136 61 37 85)(7 62 169 128 77 38 225 234 47 86 137 178 71)(8 138 217 78 63 179 48 39 170 72 87 226 129)(9 88 40 64 139 227 171 180 218 130 55 49 79)(10 181 156 29 114 96 199 208 105 123 20 147 190)(11 21 209 115 182 148 106 97 157 191 124 200 30)(12 125 98 183 22 201 158 149 210 31 192 107 116)(13 193 150 23 126 108 211 202 99 117 32 159 184)(14 33 203 109 194 160 100 91 151 185 118 212 24)(15 119 92 195 34 213 152 161 204 25 186 101 110)(16 187 162 35 120 102 205 214 93 111 26 153 196)(17 27 215 121 188 154 94 103 145 197 112 206 36)(18 113 104 189 28 207 146 155 216 19 198 95 122)
(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)
G:=sub<Sym(234)| (1,18)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,212)(38,213)(39,214)(40,215)(41,216)(42,199)(43,200)(44,201)(45,202)(46,203)(47,204)(48,205)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,109)(71,110)(72,111)(91,168)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(99,176)(100,177)(101,178)(102,179)(103,180)(104,163)(105,164)(106,165)(107,166)(108,167)(127,194)(128,195)(129,196)(130,197)(131,198)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,193)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233)(161,234)(162,217), (1,56,163,140,89,50,219,228,41,80,131,172,65)(2,132,229,90,57,173,42,51,164,66,81,220,141)(3,82,52,58,133,221,165,174,230,142,67,43,73)(4,68,175,134,83,44,231,222,53,74,143,166,59)(5,144,223,84,69,167,54,45,176,60,75,232,135)(6,76,46,70,127,233,177,168,224,136,61,37,85)(7,62,169,128,77,38,225,234,47,86,137,178,71)(8,138,217,78,63,179,48,39,170,72,87,226,129)(9,88,40,64,139,227,171,180,218,130,55,49,79)(10,181,156,29,114,96,199,208,105,123,20,147,190)(11,21,209,115,182,148,106,97,157,191,124,200,30)(12,125,98,183,22,201,158,149,210,31,192,107,116)(13,193,150,23,126,108,211,202,99,117,32,159,184)(14,33,203,109,194,160,100,91,151,185,118,212,24)(15,119,92,195,34,213,152,161,204,25,186,101,110)(16,187,162,35,120,102,205,214,93,111,26,153,196)(17,27,215,121,188,154,94,103,145,197,112,206,36)(18,113,104,189,28,207,146,155,216,19,198,95,122), (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)>;
G:=Group( (1,18)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,212)(38,213)(39,214)(40,215)(41,216)(42,199)(43,200)(44,201)(45,202)(46,203)(47,204)(48,205)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,109)(71,110)(72,111)(91,168)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(99,176)(100,177)(101,178)(102,179)(103,180)(104,163)(105,164)(106,165)(107,166)(108,167)(127,194)(128,195)(129,196)(130,197)(131,198)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,193)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,229)(157,230)(158,231)(159,232)(160,233)(161,234)(162,217), (1,56,163,140,89,50,219,228,41,80,131,172,65)(2,132,229,90,57,173,42,51,164,66,81,220,141)(3,82,52,58,133,221,165,174,230,142,67,43,73)(4,68,175,134,83,44,231,222,53,74,143,166,59)(5,144,223,84,69,167,54,45,176,60,75,232,135)(6,76,46,70,127,233,177,168,224,136,61,37,85)(7,62,169,128,77,38,225,234,47,86,137,178,71)(8,138,217,78,63,179,48,39,170,72,87,226,129)(9,88,40,64,139,227,171,180,218,130,55,49,79)(10,181,156,29,114,96,199,208,105,123,20,147,190)(11,21,209,115,182,148,106,97,157,191,124,200,30)(12,125,98,183,22,201,158,149,210,31,192,107,116)(13,193,150,23,126,108,211,202,99,117,32,159,184)(14,33,203,109,194,160,100,91,151,185,118,212,24)(15,119,92,195,34,213,152,161,204,25,186,101,110)(16,187,162,35,120,102,205,214,93,111,26,153,196)(17,27,215,121,188,154,94,103,145,197,112,206,36)(18,113,104,189,28,207,146,155,216,19,198,95,122), (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) );
G=PermutationGroup([[(1,18),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,212),(38,213),(39,214),(40,215),(41,216),(42,199),(43,200),(44,201),(45,202),(46,203),(47,204),(48,205),(49,206),(50,207),(51,208),(52,209),(53,210),(54,211),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(61,118),(62,119),(63,120),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,109),(71,110),(72,111),(91,168),(92,169),(93,170),(94,171),(95,172),(96,173),(97,174),(98,175),(99,176),(100,177),(101,178),(102,179),(103,180),(104,163),(105,164),(106,165),(107,166),(108,167),(127,194),(128,195),(129,196),(130,197),(131,198),(132,181),(133,182),(134,183),(135,184),(136,185),(137,186),(138,187),(139,188),(140,189),(141,190),(142,191),(143,192),(144,193),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,224),(152,225),(153,226),(154,227),(155,228),(156,229),(157,230),(158,231),(159,232),(160,233),(161,234),(162,217)], [(1,56,163,140,89,50,219,228,41,80,131,172,65),(2,132,229,90,57,173,42,51,164,66,81,220,141),(3,82,52,58,133,221,165,174,230,142,67,43,73),(4,68,175,134,83,44,231,222,53,74,143,166,59),(5,144,223,84,69,167,54,45,176,60,75,232,135),(6,76,46,70,127,233,177,168,224,136,61,37,85),(7,62,169,128,77,38,225,234,47,86,137,178,71),(8,138,217,78,63,179,48,39,170,72,87,226,129),(9,88,40,64,139,227,171,180,218,130,55,49,79),(10,181,156,29,114,96,199,208,105,123,20,147,190),(11,21,209,115,182,148,106,97,157,191,124,200,30),(12,125,98,183,22,201,158,149,210,31,192,107,116),(13,193,150,23,126,108,211,202,99,117,32,159,184),(14,33,203,109,194,160,100,91,151,185,118,212,24),(15,119,92,195,34,213,152,161,204,25,186,101,110),(16,187,162,35,120,102,205,214,93,111,26,153,196),(17,27,215,121,188,154,94,103,145,197,112,206,36),(18,113,104,189,28,207,146,155,216,19,198,95,122)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 13A | 13B | 18A | ··· | 18R | 26A | 26B | 39A | 39B | 39C | 39D | 78A | 78B | 78C | 78D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 13 | 13 | 18 | ··· | 18 | 26 | 26 | 39 | 39 | 39 | 39 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | ··· | 13 | 6 | 6 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C13⋊C6 | C2×C13⋊C6 | C13⋊C18 | C2×C13⋊C18 |
| kernel | C2×C13⋊C18 | C13⋊C18 | C2×C13⋊C9 | C6×D13 | C3×D13 | C78 | D26 | D13 | C26 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 2 | 2 | 4 | 4 |
Matrix representation of C2×C13⋊C18 ►in GL7(𝔽937)
| 936 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 275 | 935 | 276 | 935 | 275 | 936 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 936 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 480 | 17 | 450 | 504 | 654 | 121 |
| 0 | 457 | 4 | 631 | 867 | 427 | 837 |
| 0 | 91 | 657 | 738 | 566 | 645 | 740 |
| 0 | 554 | 290 | 637 | 290 | 554 | 0 |
| 0 | 816 | 23 | 712 | 114 | 262 | 197 |
| 0 | 100 | 130 | 204 | 204 | 130 | 100 |
G:=sub<GL(7,GF(937))| [936,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,275,1,0,0,0,0,0,935,0,1,0,0,0,0,276,0,0,1,0,0,0,935,0,0,0,1,0,0,275,0,0,0,0,1,0,936,0,0,0,0,0],[936,0,0,0,0,0,0,0,480,457,91,554,816,100,0,17,4,657,290,23,130,0,450,631,738,637,712,204,0,504,867,566,290,114,204,0,654,427,645,554,262,130,0,121,837,740,0,197,100] >;
C2×C13⋊C18 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes C_{18} % in TeX
G:=Group("C2xC13:C18"); // GroupNames label
G:=SmallGroup(468,8);
// by ID
G=gap.SmallGroup(468,8);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,57,10804,689]);
// Polycyclic
G:=Group<a,b,c|a^2=b^13=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
Export