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G = C3×C6×C12order 216 = 23·33

Abelian group of type [3,6,12]

direct product, abelian, monomial

Aliases: C3×C6×C12, SmallGroup(216,150)

Series: Derived Chief Lower central Upper central

C1 — C3×C6×C12
C1C2C6C3×C6C32×C6C32×C12 — C3×C6×C12
C1 — C3×C6×C12
C1 — C3×C6×C12

Generators and relations for C3×C6×C12
 G = < a,b,c | a3=b6=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 224, all normal (8 characteristic)
C1, C2, C2 [×2], C3 [×13], C4 [×2], C22, C6 [×39], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], C3×C6 [×39], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C32×C6, C32×C6 [×2], C6×C12 [×13], C32×C12 [×2], C3×C62, C3×C6×C12
Quotients: C1, C2 [×3], C3 [×13], C4 [×2], C22, C6 [×39], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], C3×C6 [×39], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C32×C6 [×3], C6×C12 [×13], C32×C12 [×2], C3×C62, C3×C6×C12

Smallest permutation representation of C3×C6×C12
Regular action on 216 points
Generators in S216
(1 128 196)(2 129 197)(3 130 198)(4 131 199)(5 132 200)(6 121 201)(7 122 202)(8 123 203)(9 124 204)(10 125 193)(11 126 194)(12 127 195)(13 65 106)(14 66 107)(15 67 108)(16 68 97)(17 69 98)(18 70 99)(19 71 100)(20 72 101)(21 61 102)(22 62 103)(23 63 104)(24 64 105)(25 212 156)(26 213 145)(27 214 146)(28 215 147)(29 216 148)(30 205 149)(31 206 150)(32 207 151)(33 208 152)(34 209 153)(35 210 154)(36 211 155)(37 170 117)(38 171 118)(39 172 119)(40 173 120)(41 174 109)(42 175 110)(43 176 111)(44 177 112)(45 178 113)(46 179 114)(47 180 115)(48 169 116)(49 188 137)(50 189 138)(51 190 139)(52 191 140)(53 192 141)(54 181 142)(55 182 143)(56 183 144)(57 184 133)(58 185 134)(59 186 135)(60 187 136)(73 161 88)(74 162 89)(75 163 90)(76 164 91)(77 165 92)(78 166 93)(79 167 94)(80 168 95)(81 157 96)(82 158 85)(83 159 86)(84 160 87)
(1 152 191 41 73 106)(2 153 192 42 74 107)(3 154 181 43 75 108)(4 155 182 44 76 97)(5 156 183 45 77 98)(6 145 184 46 78 99)(7 146 185 47 79 100)(8 147 186 48 80 101)(9 148 187 37 81 102)(10 149 188 38 82 103)(11 150 189 39 83 104)(12 151 190 40 84 105)(13 128 33 140 174 161)(14 129 34 141 175 162)(15 130 35 142 176 163)(16 131 36 143 177 164)(17 132 25 144 178 165)(18 121 26 133 179 166)(19 122 27 134 180 167)(20 123 28 135 169 168)(21 124 29 136 170 157)(22 125 30 137 171 158)(23 126 31 138 172 159)(24 127 32 139 173 160)(49 118 85 62 193 205)(50 119 86 63 194 206)(51 120 87 64 195 207)(52 109 88 65 196 208)(53 110 89 66 197 209)(54 111 90 67 198 210)(55 112 91 68 199 211)(56 113 92 69 200 212)(57 114 93 70 201 213)(58 115 94 71 202 214)(59 116 95 72 203 215)(60 117 96 61 204 216)
(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)| (1,128,196)(2,129,197)(3,130,198)(4,131,199)(5,132,200)(6,121,201)(7,122,202)(8,123,203)(9,124,204)(10,125,193)(11,126,194)(12,127,195)(13,65,106)(14,66,107)(15,67,108)(16,68,97)(17,69,98)(18,70,99)(19,71,100)(20,72,101)(21,61,102)(22,62,103)(23,63,104)(24,64,105)(25,212,156)(26,213,145)(27,214,146)(28,215,147)(29,216,148)(30,205,149)(31,206,150)(32,207,151)(33,208,152)(34,209,153)(35,210,154)(36,211,155)(37,170,117)(38,171,118)(39,172,119)(40,173,120)(41,174,109)(42,175,110)(43,176,111)(44,177,112)(45,178,113)(46,179,114)(47,180,115)(48,169,116)(49,188,137)(50,189,138)(51,190,139)(52,191,140)(53,192,141)(54,181,142)(55,182,143)(56,183,144)(57,184,133)(58,185,134)(59,186,135)(60,187,136)(73,161,88)(74,162,89)(75,163,90)(76,164,91)(77,165,92)(78,166,93)(79,167,94)(80,168,95)(81,157,96)(82,158,85)(83,159,86)(84,160,87), (1,152,191,41,73,106)(2,153,192,42,74,107)(3,154,181,43,75,108)(4,155,182,44,76,97)(5,156,183,45,77,98)(6,145,184,46,78,99)(7,146,185,47,79,100)(8,147,186,48,80,101)(9,148,187,37,81,102)(10,149,188,38,82,103)(11,150,189,39,83,104)(12,151,190,40,84,105)(13,128,33,140,174,161)(14,129,34,141,175,162)(15,130,35,142,176,163)(16,131,36,143,177,164)(17,132,25,144,178,165)(18,121,26,133,179,166)(19,122,27,134,180,167)(20,123,28,135,169,168)(21,124,29,136,170,157)(22,125,30,137,171,158)(23,126,31,138,172,159)(24,127,32,139,173,160)(49,118,85,62,193,205)(50,119,86,63,194,206)(51,120,87,64,195,207)(52,109,88,65,196,208)(53,110,89,66,197,209)(54,111,90,67,198,210)(55,112,91,68,199,211)(56,113,92,69,200,212)(57,114,93,70,201,213)(58,115,94,71,202,214)(59,116,95,72,203,215)(60,117,96,61,204,216), (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( (1,128,196)(2,129,197)(3,130,198)(4,131,199)(5,132,200)(6,121,201)(7,122,202)(8,123,203)(9,124,204)(10,125,193)(11,126,194)(12,127,195)(13,65,106)(14,66,107)(15,67,108)(16,68,97)(17,69,98)(18,70,99)(19,71,100)(20,72,101)(21,61,102)(22,62,103)(23,63,104)(24,64,105)(25,212,156)(26,213,145)(27,214,146)(28,215,147)(29,216,148)(30,205,149)(31,206,150)(32,207,151)(33,208,152)(34,209,153)(35,210,154)(36,211,155)(37,170,117)(38,171,118)(39,172,119)(40,173,120)(41,174,109)(42,175,110)(43,176,111)(44,177,112)(45,178,113)(46,179,114)(47,180,115)(48,169,116)(49,188,137)(50,189,138)(51,190,139)(52,191,140)(53,192,141)(54,181,142)(55,182,143)(56,183,144)(57,184,133)(58,185,134)(59,186,135)(60,187,136)(73,161,88)(74,162,89)(75,163,90)(76,164,91)(77,165,92)(78,166,93)(79,167,94)(80,168,95)(81,157,96)(82,158,85)(83,159,86)(84,160,87), (1,152,191,41,73,106)(2,153,192,42,74,107)(3,154,181,43,75,108)(4,155,182,44,76,97)(5,156,183,45,77,98)(6,145,184,46,78,99)(7,146,185,47,79,100)(8,147,186,48,80,101)(9,148,187,37,81,102)(10,149,188,38,82,103)(11,150,189,39,83,104)(12,151,190,40,84,105)(13,128,33,140,174,161)(14,129,34,141,175,162)(15,130,35,142,176,163)(16,131,36,143,177,164)(17,132,25,144,178,165)(18,121,26,133,179,166)(19,122,27,134,180,167)(20,123,28,135,169,168)(21,124,29,136,170,157)(22,125,30,137,171,158)(23,126,31,138,172,159)(24,127,32,139,173,160)(49,118,85,62,193,205)(50,119,86,63,194,206)(51,120,87,64,195,207)(52,109,88,65,196,208)(53,110,89,66,197,209)(54,111,90,67,198,210)(55,112,91,68,199,211)(56,113,92,69,200,212)(57,114,93,70,201,213)(58,115,94,71,202,214)(59,116,95,72,203,215)(60,117,96,61,204,216), (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([(1,128,196),(2,129,197),(3,130,198),(4,131,199),(5,132,200),(6,121,201),(7,122,202),(8,123,203),(9,124,204),(10,125,193),(11,126,194),(12,127,195),(13,65,106),(14,66,107),(15,67,108),(16,68,97),(17,69,98),(18,70,99),(19,71,100),(20,72,101),(21,61,102),(22,62,103),(23,63,104),(24,64,105),(25,212,156),(26,213,145),(27,214,146),(28,215,147),(29,216,148),(30,205,149),(31,206,150),(32,207,151),(33,208,152),(34,209,153),(35,210,154),(36,211,155),(37,170,117),(38,171,118),(39,172,119),(40,173,120),(41,174,109),(42,175,110),(43,176,111),(44,177,112),(45,178,113),(46,179,114),(47,180,115),(48,169,116),(49,188,137),(50,189,138),(51,190,139),(52,191,140),(53,192,141),(54,181,142),(55,182,143),(56,183,144),(57,184,133),(58,185,134),(59,186,135),(60,187,136),(73,161,88),(74,162,89),(75,163,90),(76,164,91),(77,165,92),(78,166,93),(79,167,94),(80,168,95),(81,157,96),(82,158,85),(83,159,86),(84,160,87)], [(1,152,191,41,73,106),(2,153,192,42,74,107),(3,154,181,43,75,108),(4,155,182,44,76,97),(5,156,183,45,77,98),(6,145,184,46,78,99),(7,146,185,47,79,100),(8,147,186,48,80,101),(9,148,187,37,81,102),(10,149,188,38,82,103),(11,150,189,39,83,104),(12,151,190,40,84,105),(13,128,33,140,174,161),(14,129,34,141,175,162),(15,130,35,142,176,163),(16,131,36,143,177,164),(17,132,25,144,178,165),(18,121,26,133,179,166),(19,122,27,134,180,167),(20,123,28,135,169,168),(21,124,29,136,170,157),(22,125,30,137,171,158),(23,126,31,138,172,159),(24,127,32,139,173,160),(49,118,85,62,193,205),(50,119,86,63,194,206),(51,120,87,64,195,207),(52,109,88,65,196,208),(53,110,89,66,197,209),(54,111,90,67,198,210),(55,112,91,68,199,211),(56,113,92,69,200,212),(57,114,93,70,201,213),(58,115,94,71,202,214),(59,116,95,72,203,215),(60,117,96,61,204,216)], [(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)])

C3×C6×C12 is a maximal subgroup of   C3318M4(2)  C62.146D6  C62.147D6  C62.148D6  C62.160D6

216 conjugacy classes

class 1 2A2B2C3A···3Z4A4B4C4D6A···6BZ12A···12CZ
order12223···344446···612···12
size11111···111111···11···1

216 irreducible representations

dim11111111
type+++
imageC1C2C2C3C4C6C6C12
kernelC3×C6×C12C32×C12C3×C62C6×C12C32×C6C3×C12C62C3×C6
# reps1212645226104

Matrix representation of C3×C6×C12 in GL4(𝔽13) generated by

1000
0300
0090
0009
,
12000
0400
00100
0001
,
1000
0700
0060
00012
G:=sub<GL(4,GF(13))| [1,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,4,0,0,0,0,10,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,6,0,0,0,0,12] >;

C3×C6×C12 in GAP, Magma, Sage, TeX

C_3\times C_6\times C_{12}
% in TeX

G:=Group("C3xC6xC12");
// GroupNames label

G:=SmallGroup(216,150);
// by ID

G=gap.SmallGroup(216,150);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-2,648]);
// Polycyclic

G:=Group<a,b,c|a^3=b^6=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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