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## G = C3×C6×C12order 216 = 23·33

### Abelian group of type [3,6,12]

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6×C12
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C32×C12 — C3×C6×C12
 Lower central C1 — C3×C6×C12
 Upper central 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, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, C2×C12, C33, C3×C12, C62, C32×C6, C32×C6, C6×C12, C32×C12, C3×C62, C3×C6×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, C2×C12, C33, C3×C12, C62, C32×C6, C6×C12, C32×C12, C3×C62, C3×C6×C12

Smallest permutation representation of C3×C6×C12
Regular action on 216 points
Generators in S216
(1 131 205)(2 132 206)(3 121 207)(4 122 208)(5 123 209)(6 124 210)(7 125 211)(8 126 212)(9 127 213)(10 128 214)(11 129 215)(12 130 216)(13 65 82)(14 66 83)(15 67 84)(16 68 73)(17 69 74)(18 70 75)(19 71 76)(20 72 77)(21 61 78)(22 62 79)(23 63 80)(24 64 81)(25 93 109)(26 94 110)(27 95 111)(28 96 112)(29 85 113)(30 86 114)(31 87 115)(32 88 116)(33 89 117)(34 90 118)(35 91 119)(36 92 120)(37 174 160)(38 175 161)(39 176 162)(40 177 163)(41 178 164)(42 179 165)(43 180 166)(44 169 167)(45 170 168)(46 171 157)(47 172 158)(48 173 159)(49 194 191)(50 195 192)(51 196 181)(52 197 182)(53 198 183)(54 199 184)(55 200 185)(56 201 186)(57 202 187)(58 203 188)(59 204 189)(60 193 190)(97 155 139)(98 156 140)(99 145 141)(100 146 142)(101 147 143)(102 148 144)(103 149 133)(104 150 134)(105 151 135)(106 152 136)(107 153 137)(108 154 138)
(1 34 133 24 164 54)(2 35 134 13 165 55)(3 36 135 14 166 56)(4 25 136 15 167 57)(5 26 137 16 168 58)(6 27 138 17 157 59)(7 28 139 18 158 60)(8 29 140 19 159 49)(9 30 141 20 160 50)(10 31 142 21 161 51)(11 32 143 22 162 52)(12 33 144 23 163 53)(37 195 127 86 99 72)(38 196 128 87 100 61)(39 197 129 88 101 62)(40 198 130 89 102 63)(41 199 131 90 103 64)(42 200 132 91 104 65)(43 201 121 92 105 66)(44 202 122 93 106 67)(45 203 123 94 107 68)(46 204 124 95 108 69)(47 193 125 96 97 70)(48 194 126 85 98 71)(73 170 188 209 110 153)(74 171 189 210 111 154)(75 172 190 211 112 155)(76 173 191 212 113 156)(77 174 192 213 114 145)(78 175 181 214 115 146)(79 176 182 215 116 147)(80 177 183 216 117 148)(81 178 184 205 118 149)(82 179 185 206 119 150)(83 180 186 207 120 151)(84 169 187 208 109 152)
(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)| (1,131,205)(2,132,206)(3,121,207)(4,122,208)(5,123,209)(6,124,210)(7,125,211)(8,126,212)(9,127,213)(10,128,214)(11,129,215)(12,130,216)(13,65,82)(14,66,83)(15,67,84)(16,68,73)(17,69,74)(18,70,75)(19,71,76)(20,72,77)(21,61,78)(22,62,79)(23,63,80)(24,64,81)(25,93,109)(26,94,110)(27,95,111)(28,96,112)(29,85,113)(30,86,114)(31,87,115)(32,88,116)(33,89,117)(34,90,118)(35,91,119)(36,92,120)(37,174,160)(38,175,161)(39,176,162)(40,177,163)(41,178,164)(42,179,165)(43,180,166)(44,169,167)(45,170,168)(46,171,157)(47,172,158)(48,173,159)(49,194,191)(50,195,192)(51,196,181)(52,197,182)(53,198,183)(54,199,184)(55,200,185)(56,201,186)(57,202,187)(58,203,188)(59,204,189)(60,193,190)(97,155,139)(98,156,140)(99,145,141)(100,146,142)(101,147,143)(102,148,144)(103,149,133)(104,150,134)(105,151,135)(106,152,136)(107,153,137)(108,154,138), (1,34,133,24,164,54)(2,35,134,13,165,55)(3,36,135,14,166,56)(4,25,136,15,167,57)(5,26,137,16,168,58)(6,27,138,17,157,59)(7,28,139,18,158,60)(8,29,140,19,159,49)(9,30,141,20,160,50)(10,31,142,21,161,51)(11,32,143,22,162,52)(12,33,144,23,163,53)(37,195,127,86,99,72)(38,196,128,87,100,61)(39,197,129,88,101,62)(40,198,130,89,102,63)(41,199,131,90,103,64)(42,200,132,91,104,65)(43,201,121,92,105,66)(44,202,122,93,106,67)(45,203,123,94,107,68)(46,204,124,95,108,69)(47,193,125,96,97,70)(48,194,126,85,98,71)(73,170,188,209,110,153)(74,171,189,210,111,154)(75,172,190,211,112,155)(76,173,191,212,113,156)(77,174,192,213,114,145)(78,175,181,214,115,146)(79,176,182,215,116,147)(80,177,183,216,117,148)(81,178,184,205,118,149)(82,179,185,206,119,150)(83,180,186,207,120,151)(84,169,187,208,109,152), (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( (1,131,205)(2,132,206)(3,121,207)(4,122,208)(5,123,209)(6,124,210)(7,125,211)(8,126,212)(9,127,213)(10,128,214)(11,129,215)(12,130,216)(13,65,82)(14,66,83)(15,67,84)(16,68,73)(17,69,74)(18,70,75)(19,71,76)(20,72,77)(21,61,78)(22,62,79)(23,63,80)(24,64,81)(25,93,109)(26,94,110)(27,95,111)(28,96,112)(29,85,113)(30,86,114)(31,87,115)(32,88,116)(33,89,117)(34,90,118)(35,91,119)(36,92,120)(37,174,160)(38,175,161)(39,176,162)(40,177,163)(41,178,164)(42,179,165)(43,180,166)(44,169,167)(45,170,168)(46,171,157)(47,172,158)(48,173,159)(49,194,191)(50,195,192)(51,196,181)(52,197,182)(53,198,183)(54,199,184)(55,200,185)(56,201,186)(57,202,187)(58,203,188)(59,204,189)(60,193,190)(97,155,139)(98,156,140)(99,145,141)(100,146,142)(101,147,143)(102,148,144)(103,149,133)(104,150,134)(105,151,135)(106,152,136)(107,153,137)(108,154,138), (1,34,133,24,164,54)(2,35,134,13,165,55)(3,36,135,14,166,56)(4,25,136,15,167,57)(5,26,137,16,168,58)(6,27,138,17,157,59)(7,28,139,18,158,60)(8,29,140,19,159,49)(9,30,141,20,160,50)(10,31,142,21,161,51)(11,32,143,22,162,52)(12,33,144,23,163,53)(37,195,127,86,99,72)(38,196,128,87,100,61)(39,197,129,88,101,62)(40,198,130,89,102,63)(41,199,131,90,103,64)(42,200,132,91,104,65)(43,201,121,92,105,66)(44,202,122,93,106,67)(45,203,123,94,107,68)(46,204,124,95,108,69)(47,193,125,96,97,70)(48,194,126,85,98,71)(73,170,188,209,110,153)(74,171,189,210,111,154)(75,172,190,211,112,155)(76,173,191,212,113,156)(77,174,192,213,114,145)(78,175,181,214,115,146)(79,176,182,215,116,147)(80,177,183,216,117,148)(81,178,184,205,118,149)(82,179,185,206,119,150)(83,180,186,207,120,151)(84,169,187,208,109,152), (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([[(1,131,205),(2,132,206),(3,121,207),(4,122,208),(5,123,209),(6,124,210),(7,125,211),(8,126,212),(9,127,213),(10,128,214),(11,129,215),(12,130,216),(13,65,82),(14,66,83),(15,67,84),(16,68,73),(17,69,74),(18,70,75),(19,71,76),(20,72,77),(21,61,78),(22,62,79),(23,63,80),(24,64,81),(25,93,109),(26,94,110),(27,95,111),(28,96,112),(29,85,113),(30,86,114),(31,87,115),(32,88,116),(33,89,117),(34,90,118),(35,91,119),(36,92,120),(37,174,160),(38,175,161),(39,176,162),(40,177,163),(41,178,164),(42,179,165),(43,180,166),(44,169,167),(45,170,168),(46,171,157),(47,172,158),(48,173,159),(49,194,191),(50,195,192),(51,196,181),(52,197,182),(53,198,183),(54,199,184),(55,200,185),(56,201,186),(57,202,187),(58,203,188),(59,204,189),(60,193,190),(97,155,139),(98,156,140),(99,145,141),(100,146,142),(101,147,143),(102,148,144),(103,149,133),(104,150,134),(105,151,135),(106,152,136),(107,153,137),(108,154,138)], [(1,34,133,24,164,54),(2,35,134,13,165,55),(3,36,135,14,166,56),(4,25,136,15,167,57),(5,26,137,16,168,58),(6,27,138,17,157,59),(7,28,139,18,158,60),(8,29,140,19,159,49),(9,30,141,20,160,50),(10,31,142,21,161,51),(11,32,143,22,162,52),(12,33,144,23,163,53),(37,195,127,86,99,72),(38,196,128,87,100,61),(39,197,129,88,101,62),(40,198,130,89,102,63),(41,199,131,90,103,64),(42,200,132,91,104,65),(43,201,121,92,105,66),(44,202,122,93,106,67),(45,203,123,94,107,68),(46,204,124,95,108,69),(47,193,125,96,97,70),(48,194,126,85,98,71),(73,170,188,209,110,153),(74,171,189,210,111,154),(75,172,190,211,112,155),(76,173,191,212,113,156),(77,174,192,213,114,145),(78,175,181,214,115,146),(79,176,182,215,116,147),(80,177,183,216,117,148),(81,178,184,205,118,149),(82,179,185,206,119,150),(83,180,186,207,120,151),(84,169,187,208,109,152)], [(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 2A 2B 2C 3A ··· 3Z 4A 4B 4C 4D 6A ··· 6BZ 12A ··· 12CZ order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 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 C2 C3 C4 C6 C6 C12 kernel C3×C6×C12 C32×C12 C3×C62 C6×C12 C32×C6 C3×C12 C62 C3×C6 # reps 1 2 1 26 4 52 26 104

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

 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
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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