direct product, abelian, monomial
Aliases: C2×C6×C18, SmallGroup(216,114)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C6×C18 |
| C1 — C2×C6×C18 |
| C1 — C2×C6×C18 |
Generators and relations for C2×C6×C18
G = < a,b,c | a2=b6=c18=1, ab=ba, ac=ca, bc=cb >
Subgroups: 160, all normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C23, C9, C32, C2×C6, C18, C3×C6, C22×C6, C22×C6, C3×C9, C2×C18, C62, C3×C18, C22×C18, C2×C62, C6×C18, C2×C6×C18
Quotients: C1, C2, C3, C22, C6, C23, C9, C32, C2×C6, C18, C3×C6, C22×C6, C3×C9, C2×C18, C62, C3×C18, C22×C18, C2×C62, C6×C18, C2×C6×C18
►Regular action on 216 points
Generators in S216
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 197)(20 198)(21 181)(22 182)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 109)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 145)(143 146)(144 147)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)(169 209)(170 210)(171 211)(172 212)(173 213)(174 214)(175 215)(176 216)(177 199)(178 200)(179 201)(180 202)
(1 201 102 38 187 133)(2 202 103 39 188 134)(3 203 104 40 189 135)(4 204 105 41 190 136)(5 205 106 42 191 137)(6 206 107 43 192 138)(7 207 108 44 193 139)(8 208 91 45 194 140)(9 209 92 46 195 141)(10 210 93 47 196 142)(11 211 94 48 197 143)(12 212 95 49 198 144)(13 213 96 50 181 127)(14 214 97 51 182 128)(15 215 98 52 183 129)(16 216 99 53 184 130)(17 199 100 54 185 131)(18 200 101 37 186 132)(19 146 57 171 82 121)(20 147 58 172 83 122)(21 148 59 173 84 123)(22 149 60 174 85 124)(23 150 61 175 86 125)(24 151 62 176 87 126)(25 152 63 177 88 109)(26 153 64 178 89 110)(27 154 65 179 90 111)(28 155 66 180 73 112)(29 156 67 163 74 113)(30 157 68 164 75 114)(31 158 69 165 76 115)(32 159 70 166 77 116)(33 160 71 167 78 117)(34 161 72 168 79 118)(35 162 55 169 80 119)(36 145 56 170 81 120)
(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)
G:=sub<Sym(216)| (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,197)(20,198)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,109)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,145)(143,146)(144,147)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,199)(178,200)(179,201)(180,202), (1,201,102,38,187,133)(2,202,103,39,188,134)(3,203,104,40,189,135)(4,204,105,41,190,136)(5,205,106,42,191,137)(6,206,107,43,192,138)(7,207,108,44,193,139)(8,208,91,45,194,140)(9,209,92,46,195,141)(10,210,93,47,196,142)(11,211,94,48,197,143)(12,212,95,49,198,144)(13,213,96,50,181,127)(14,214,97,51,182,128)(15,215,98,52,183,129)(16,216,99,53,184,130)(17,199,100,54,185,131)(18,200,101,37,186,132)(19,146,57,171,82,121)(20,147,58,172,83,122)(21,148,59,173,84,123)(22,149,60,174,85,124)(23,150,61,175,86,125)(24,151,62,176,87,126)(25,152,63,177,88,109)(26,153,64,178,89,110)(27,154,65,179,90,111)(28,155,66,180,73,112)(29,156,67,163,74,113)(30,157,68,164,75,114)(31,158,69,165,76,115)(32,159,70,166,77,116)(33,160,71,167,78,117)(34,161,72,168,79,118)(35,162,55,169,80,119)(36,145,56,170,81,120), (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)>;
G:=Group( (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,197)(20,198)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,109)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,145)(143,146)(144,147)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,199)(178,200)(179,201)(180,202), (1,201,102,38,187,133)(2,202,103,39,188,134)(3,203,104,40,189,135)(4,204,105,41,190,136)(5,205,106,42,191,137)(6,206,107,43,192,138)(7,207,108,44,193,139)(8,208,91,45,194,140)(9,209,92,46,195,141)(10,210,93,47,196,142)(11,211,94,48,197,143)(12,212,95,49,198,144)(13,213,96,50,181,127)(14,214,97,51,182,128)(15,215,98,52,183,129)(16,216,99,53,184,130)(17,199,100,54,185,131)(18,200,101,37,186,132)(19,146,57,171,82,121)(20,147,58,172,83,122)(21,148,59,173,84,123)(22,149,60,174,85,124)(23,150,61,175,86,125)(24,151,62,176,87,126)(25,152,63,177,88,109)(26,153,64,178,89,110)(27,154,65,179,90,111)(28,155,66,180,73,112)(29,156,67,163,74,113)(30,157,68,164,75,114)(31,158,69,165,76,115)(32,159,70,166,77,116)(33,160,71,167,78,117)(34,161,72,168,79,118)(35,162,55,169,80,119)(36,145,56,170,81,120), (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) );
G=PermutationGroup([[(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,197),(20,198),(21,181),(22,182),(23,183),(24,184),(25,185),(26,186),(27,187),(28,188),(29,189),(30,190),(31,191),(32,192),(33,193),(34,194),(35,195),(36,196),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,109),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,145),(143,146),(144,147),(163,203),(164,204),(165,205),(166,206),(167,207),(168,208),(169,209),(170,210),(171,211),(172,212),(173,213),(174,214),(175,215),(176,216),(177,199),(178,200),(179,201),(180,202)], [(1,201,102,38,187,133),(2,202,103,39,188,134),(3,203,104,40,189,135),(4,204,105,41,190,136),(5,205,106,42,191,137),(6,206,107,43,192,138),(7,207,108,44,193,139),(8,208,91,45,194,140),(9,209,92,46,195,141),(10,210,93,47,196,142),(11,211,94,48,197,143),(12,212,95,49,198,144),(13,213,96,50,181,127),(14,214,97,51,182,128),(15,215,98,52,183,129),(16,216,99,53,184,130),(17,199,100,54,185,131),(18,200,101,37,186,132),(19,146,57,171,82,121),(20,147,58,172,83,122),(21,148,59,173,84,123),(22,149,60,174,85,124),(23,150,61,175,86,125),(24,151,62,176,87,126),(25,152,63,177,88,109),(26,153,64,178,89,110),(27,154,65,179,90,111),(28,155,66,180,73,112),(29,156,67,163,74,113),(30,157,68,164,75,114),(31,158,69,165,76,115),(32,159,70,166,77,116),(33,160,71,167,78,117),(34,161,72,168,79,118),(35,162,55,169,80,119),(36,145,56,170,81,120)], [(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)]])
C2×C6×C18 is a maximal subgroup of
C62.127D6
216 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3H | 6A | ··· | 6BD | 9A | ··· | 9R | 18A | ··· | 18DV |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 |
| kernel | C2×C6×C18 | C6×C18 | C22×C18 | C2×C62 | C2×C18 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 7 | 6 | 2 | 42 | 14 | 18 | 126 |
Matrix representation of C2×C6×C18 ►in GL3(𝔽19) generated by
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 18 |
| 14 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(19))| [1,0,0,0,18,0,0,0,18],[8,0,0,0,8,0,0,0,18],[14,0,0,0,14,0,0,0,6] >;
C2×C6×C18 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_{18} % in TeX
G:=Group("C2xC6xC18"); // GroupNames label
G:=SmallGroup(216,114);
// by ID
G=gap.SmallGroup(216,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,237]);
// Polycyclic
G:=Group<a,b,c|a^2=b^6=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations