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G = C2×C6×C18order 216 = 23·33

Abelian group of type [2,6,18]

direct product, abelian, monomial

Aliases: C2×C6×C18, SmallGroup(216,114)

Series: Derived Chief Lower central Upper central

C1 — C2×C6×C18
C1C3C32C3×C9C3×C18C6×C18 — 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 [×7], C3, C3 [×3], C22 [×7], C6 [×28], C23, C9 [×3], C32, C2×C6 [×28], C18 [×21], C3×C6 [×7], C22×C6, C22×C6 [×3], C3×C9, C2×C18 [×21], C62 [×7], C3×C18 [×7], C22×C18 [×3], C2×C62, C6×C18 [×7], C2×C6×C18
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], C23, C9 [×3], C32, C2×C6 [×28], C18 [×21], C3×C6 [×7], C22×C6 [×4], C3×C9, C2×C18 [×21], C62 [×7], C3×C18 [×7], C22×C18 [×3], C2×C62, C6×C18 [×7], C2×C6×C18

Smallest permutation representation of C2×C6×C18
Regular action on 216 points
Generators in S216
(1 106)(2 107)(3 108)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 73)(36 74)(37 130)(38 131)(39 132)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 144)(52 127)(53 128)(54 129)(55 160)(56 161)(57 162)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(109 179)(110 180)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(181 214)(182 215)(183 216)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(196 211)(197 212)(198 213)
(1 155 87 53 190 172)(2 156 88 54 191 173)(3 157 89 37 192 174)(4 158 90 38 193 175)(5 159 73 39 194 176)(6 160 74 40 195 177)(7 161 75 41 196 178)(8 162 76 42 197 179)(9 145 77 43 198 180)(10 146 78 44 181 163)(11 147 79 45 182 164)(12 148 80 46 183 165)(13 149 81 47 184 166)(14 150 82 48 185 167)(15 151 83 49 186 168)(16 152 84 50 187 169)(17 153 85 51 188 170)(18 154 86 52 189 171)(19 134 211 126 94 56)(20 135 212 109 95 57)(21 136 213 110 96 58)(22 137 214 111 97 59)(23 138 215 112 98 60)(24 139 216 113 99 61)(25 140 199 114 100 62)(26 141 200 115 101 63)(27 142 201 116 102 64)(28 143 202 117 103 65)(29 144 203 118 104 66)(30 127 204 119 105 67)(31 128 205 120 106 68)(32 129 206 121 107 69)(33 130 207 122 108 70)(34 131 208 123 91 71)(35 132 209 124 92 72)(36 133 210 125 93 55)
(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)| (1,106)(2,107)(3,108)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,73)(36,74)(37,130)(38,131)(39,132)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,127)(53,128)(54,129)(55,160)(56,161)(57,162)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(109,179)(110,180)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(181,214)(182,215)(183,216)(184,199)(185,200)(186,201)(187,202)(188,203)(189,204)(190,205)(191,206)(192,207)(193,208)(194,209)(195,210)(196,211)(197,212)(198,213), (1,155,87,53,190,172)(2,156,88,54,191,173)(3,157,89,37,192,174)(4,158,90,38,193,175)(5,159,73,39,194,176)(6,160,74,40,195,177)(7,161,75,41,196,178)(8,162,76,42,197,179)(9,145,77,43,198,180)(10,146,78,44,181,163)(11,147,79,45,182,164)(12,148,80,46,183,165)(13,149,81,47,184,166)(14,150,82,48,185,167)(15,151,83,49,186,168)(16,152,84,50,187,169)(17,153,85,51,188,170)(18,154,86,52,189,171)(19,134,211,126,94,56)(20,135,212,109,95,57)(21,136,213,110,96,58)(22,137,214,111,97,59)(23,138,215,112,98,60)(24,139,216,113,99,61)(25,140,199,114,100,62)(26,141,200,115,101,63)(27,142,201,116,102,64)(28,143,202,117,103,65)(29,144,203,118,104,66)(30,127,204,119,105,67)(31,128,205,120,106,68)(32,129,206,121,107,69)(33,130,207,122,108,70)(34,131,208,123,91,71)(35,132,209,124,92,72)(36,133,210,125,93,55), (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( (1,106)(2,107)(3,108)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,73)(36,74)(37,130)(38,131)(39,132)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,127)(53,128)(54,129)(55,160)(56,161)(57,162)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(109,179)(110,180)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(181,214)(182,215)(183,216)(184,199)(185,200)(186,201)(187,202)(188,203)(189,204)(190,205)(191,206)(192,207)(193,208)(194,209)(195,210)(196,211)(197,212)(198,213), (1,155,87,53,190,172)(2,156,88,54,191,173)(3,157,89,37,192,174)(4,158,90,38,193,175)(5,159,73,39,194,176)(6,160,74,40,195,177)(7,161,75,41,196,178)(8,162,76,42,197,179)(9,145,77,43,198,180)(10,146,78,44,181,163)(11,147,79,45,182,164)(12,148,80,46,183,165)(13,149,81,47,184,166)(14,150,82,48,185,167)(15,151,83,49,186,168)(16,152,84,50,187,169)(17,153,85,51,188,170)(18,154,86,52,189,171)(19,134,211,126,94,56)(20,135,212,109,95,57)(21,136,213,110,96,58)(22,137,214,111,97,59)(23,138,215,112,98,60)(24,139,216,113,99,61)(25,140,199,114,100,62)(26,141,200,115,101,63)(27,142,201,116,102,64)(28,143,202,117,103,65)(29,144,203,118,104,66)(30,127,204,119,105,67)(31,128,205,120,106,68)(32,129,206,121,107,69)(33,130,207,122,108,70)(34,131,208,123,91,71)(35,132,209,124,92,72)(36,133,210,125,93,55), (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([(1,106),(2,107),(3,108),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,73),(36,74),(37,130),(38,131),(39,132),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(49,142),(50,143),(51,144),(52,127),(53,128),(54,129),(55,160),(56,161),(57,162),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,158),(72,159),(109,179),(110,180),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,169),(118,170),(119,171),(120,172),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(181,214),(182,215),(183,216),(184,199),(185,200),(186,201),(187,202),(188,203),(189,204),(190,205),(191,206),(192,207),(193,208),(194,209),(195,210),(196,211),(197,212),(198,213)], [(1,155,87,53,190,172),(2,156,88,54,191,173),(3,157,89,37,192,174),(4,158,90,38,193,175),(5,159,73,39,194,176),(6,160,74,40,195,177),(7,161,75,41,196,178),(8,162,76,42,197,179),(9,145,77,43,198,180),(10,146,78,44,181,163),(11,147,79,45,182,164),(12,148,80,46,183,165),(13,149,81,47,184,166),(14,150,82,48,185,167),(15,151,83,49,186,168),(16,152,84,50,187,169),(17,153,85,51,188,170),(18,154,86,52,189,171),(19,134,211,126,94,56),(20,135,212,109,95,57),(21,136,213,110,96,58),(22,137,214,111,97,59),(23,138,215,112,98,60),(24,139,216,113,99,61),(25,140,199,114,100,62),(26,141,200,115,101,63),(27,142,201,116,102,64),(28,143,202,117,103,65),(29,144,203,118,104,66),(30,127,204,119,105,67),(31,128,205,120,106,68),(32,129,206,121,107,69),(33,130,207,122,108,70),(34,131,208,123,91,71),(35,132,209,124,92,72),(36,133,210,125,93,55)], [(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···2G3A···3H6A···6BD9A···9R18A···18DV
order12···23···36···69···918···18
size11···11···11···11···11···1

216 irreducible representations

dim11111111
type++
imageC1C2C3C3C6C6C9C18
kernelC2×C6×C18C6×C18C22×C18C2×C62C2×C18C62C22×C6C2×C6
# reps1762421418126

Matrix representation of C2×C6×C18 in GL3(𝔽19) generated by

100
0180
0018
,
800
080
0018
,
1400
0140
006
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

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