direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C18×D11, C22⋊C18, C198⋊2C2, C66.3C6, C99⋊3C22, C11⋊(C2×C18), C33.(C2×C6), C3.(C6×D11), (C6×D11).C3, (C3×D11).C6, C6.3(C3×D11), SmallGroup(396,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C18×D11 |
Generators and relations for C18×D11
G = < a,b,c | a18=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C18×D11
►On 198 points
Generators in S198
(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)
(1 63 41 185 30 179 88 115 94 142 157)(2 64 42 186 31 180 89 116 95 143 158)(3 65 43 187 32 163 90 117 96 144 159)(4 66 44 188 33 164 73 118 97 127 160)(5 67 45 189 34 165 74 119 98 128 161)(6 68 46 190 35 166 75 120 99 129 162)(7 69 47 191 36 167 76 121 100 130 145)(8 70 48 192 19 168 77 122 101 131 146)(9 71 49 193 20 169 78 123 102 132 147)(10 72 50 194 21 170 79 124 103 133 148)(11 55 51 195 22 171 80 125 104 134 149)(12 56 52 196 23 172 81 126 105 135 150)(13 57 53 197 24 173 82 109 106 136 151)(14 58 54 198 25 174 83 110 107 137 152)(15 59 37 181 26 175 84 111 108 138 153)(16 60 38 182 27 176 85 112 91 139 154)(17 61 39 183 28 177 86 113 92 140 155)(18 62 40 184 29 178 87 114 93 141 156)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 157)(11 158)(12 159)(13 160)(14 161)(15 162)(16 145)(17 146)(18 147)(19 86)(20 87)(21 88)(22 89)(23 90)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 143)(56 144)(57 127)(58 128)(59 129)(60 130)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(109 188)(110 189)(111 190)(112 191)(113 192)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)(170 179)(171 180)
G:=sub<Sym(198)| (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), (1,63,41,185,30,179,88,115,94,142,157)(2,64,42,186,31,180,89,116,95,143,158)(3,65,43,187,32,163,90,117,96,144,159)(4,66,44,188,33,164,73,118,97,127,160)(5,67,45,189,34,165,74,119,98,128,161)(6,68,46,190,35,166,75,120,99,129,162)(7,69,47,191,36,167,76,121,100,130,145)(8,70,48,192,19,168,77,122,101,131,146)(9,71,49,193,20,169,78,123,102,132,147)(10,72,50,194,21,170,79,124,103,133,148)(11,55,51,195,22,171,80,125,104,134,149)(12,56,52,196,23,172,81,126,105,135,150)(13,57,53,197,24,173,82,109,106,136,151)(14,58,54,198,25,174,83,110,107,137,152)(15,59,37,181,26,175,84,111,108,138,153)(16,60,38,182,27,176,85,112,91,139,154)(17,61,39,183,28,177,86,113,92,140,155)(18,62,40,184,29,178,87,114,93,141,156), (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,161)(15,162)(16,145)(17,146)(18,147)(19,86)(20,87)(21,88)(22,89)(23,90)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,143)(56,144)(57,127)(58,128)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(109,188)(110,189)(111,190)(112,191)(113,192)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)>;
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), (1,63,41,185,30,179,88,115,94,142,157)(2,64,42,186,31,180,89,116,95,143,158)(3,65,43,187,32,163,90,117,96,144,159)(4,66,44,188,33,164,73,118,97,127,160)(5,67,45,189,34,165,74,119,98,128,161)(6,68,46,190,35,166,75,120,99,129,162)(7,69,47,191,36,167,76,121,100,130,145)(8,70,48,192,19,168,77,122,101,131,146)(9,71,49,193,20,169,78,123,102,132,147)(10,72,50,194,21,170,79,124,103,133,148)(11,55,51,195,22,171,80,125,104,134,149)(12,56,52,196,23,172,81,126,105,135,150)(13,57,53,197,24,173,82,109,106,136,151)(14,58,54,198,25,174,83,110,107,137,152)(15,59,37,181,26,175,84,111,108,138,153)(16,60,38,182,27,176,85,112,91,139,154)(17,61,39,183,28,177,86,113,92,140,155)(18,62,40,184,29,178,87,114,93,141,156), (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,161)(15,162)(16,145)(17,146)(18,147)(19,86)(20,87)(21,88)(22,89)(23,90)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,143)(56,144)(57,127)(58,128)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(109,188)(110,189)(111,190)(112,191)(113,192)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180) );
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)], [(1,63,41,185,30,179,88,115,94,142,157),(2,64,42,186,31,180,89,116,95,143,158),(3,65,43,187,32,163,90,117,96,144,159),(4,66,44,188,33,164,73,118,97,127,160),(5,67,45,189,34,165,74,119,98,128,161),(6,68,46,190,35,166,75,120,99,129,162),(7,69,47,191,36,167,76,121,100,130,145),(8,70,48,192,19,168,77,122,101,131,146),(9,71,49,193,20,169,78,123,102,132,147),(10,72,50,194,21,170,79,124,103,133,148),(11,55,51,195,22,171,80,125,104,134,149),(12,56,52,196,23,172,81,126,105,135,150),(13,57,53,197,24,173,82,109,106,136,151),(14,58,54,198,25,174,83,110,107,137,152),(15,59,37,181,26,175,84,111,108,138,153),(16,60,38,182,27,176,85,112,91,139,154),(17,61,39,183,28,177,86,113,92,140,155),(18,62,40,184,29,178,87,114,93,141,156)], [(1,148),(2,149),(3,150),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,157),(11,158),(12,159),(13,160),(14,161),(15,162),(16,145),(17,146),(18,147),(19,86),(20,87),(21,88),(22,89),(23,90),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,85),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,91),(48,92),(49,93),(50,94),(51,95),(52,96),(53,97),(54,98),(55,143),(56,144),(57,127),(58,128),(59,129),(60,130),(61,131),(62,132),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(109,188),(110,189),(111,190),(112,191),(113,192),(114,193),(115,194),(116,195),(117,196),(118,197),(119,198),(120,181),(121,182),(122,183),(123,184),(124,185),(125,186),(126,187),(163,172),(164,173),(165,174),(166,175),(167,176),(168,177),(169,178),(170,179),(171,180)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 11A | ··· | 11E | 18A | ··· | 18F | 18G | ··· | 18R | 22A | ··· | 22E | 33A | ··· | 33J | 66A | ··· | 66J | 99A | ··· | 99AD | 198A | ··· | 198AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 11 | ··· | 11 | 18 | ··· | 18 | 18 | ··· | 18 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 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 | D11 | D22 | C3×D11 | C6×D11 | C9×D11 | C18×D11 |
| kernel | C18×D11 | C9×D11 | C198 | C6×D11 | C3×D11 | C66 | D22 | D11 | C22 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 5 | 5 | 10 | 10 | 30 | 30 |
Matrix representation of C18×D11 ►in GL3(𝔽199) generated by
| 107 | 0 | 0 |
| 0 | 175 | 0 |
| 0 | 0 | 175 |
| 1 | 0 | 0 |
| 0 | 142 | 97 |
| 0 | 198 | 75 |
| 1 | 0 | 0 |
| 0 | 32 | 109 |
| 0 | 18 | 167 |
G:=sub<GL(3,GF(199))| [107,0,0,0,175,0,0,0,175],[1,0,0,0,142,198,0,97,75],[1,0,0,0,32,18,0,109,167] >;
C18×D11 in GAP, Magma, Sage, TeX
C_{18}\times D_{11} % in TeX
G:=Group("C18xD11"); // GroupNames label
G:=SmallGroup(396,7);
// by ID
G=gap.SmallGroup(396,7);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-11,57,9004]);
// Polycyclic
G:=Group<a,b,c|a^18=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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