direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic11, C22⋊C12, C66⋊2C4, C6.16D22, C66.16C22, C33⋊8(C2×C4), (C2×C22).C6, C11⋊2(C2×C12), (C2×C66).2C2, C22.4(C2×C6), (C2×C6).2D11, C2.2(C6×D11), C22.(C3×D11), SmallGroup(264,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C6×Dic11 |
Generators and relations for C6×Dic11
G = < a,b,c | a6=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic11
►Regular action on 264 points
Generators in S264
(1 96 62 70 23 115)(2 97 63 71 24 116)(3 98 64 72 25 117)(4 99 65 73 26 118)(5 100 66 74 27 119)(6 101 45 75 28 120)(7 102 46 76 29 121)(8 103 47 77 30 122)(9 104 48 78 31 123)(10 105 49 79 32 124)(11 106 50 80 33 125)(12 107 51 81 34 126)(13 108 52 82 35 127)(14 109 53 83 36 128)(15 110 54 84 37 129)(16 89 55 85 38 130)(17 90 56 86 39 131)(18 91 57 87 40 132)(19 92 58 88 41 111)(20 93 59 67 42 112)(21 94 60 68 43 113)(22 95 61 69 44 114)(133 232 177 199 155 254)(134 233 178 200 156 255)(135 234 179 201 157 256)(136 235 180 202 158 257)(137 236 181 203 159 258)(138 237 182 204 160 259)(139 238 183 205 161 260)(140 239 184 206 162 261)(141 240 185 207 163 262)(142 241 186 208 164 263)(143 242 187 209 165 264)(144 221 188 210 166 243)(145 222 189 211 167 244)(146 223 190 212 168 245)(147 224 191 213 169 246)(148 225 192 214 170 247)(149 226 193 215 171 248)(150 227 194 216 172 249)(151 228 195 217 173 250)(152 229 196 218 174 251)(153 230 197 219 175 252)(154 231 198 220 176 253)
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242)(243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264)
(1 144 12 133)(2 143 13 154)(3 142 14 153)(4 141 15 152)(5 140 16 151)(6 139 17 150)(7 138 18 149)(8 137 19 148)(9 136 20 147)(10 135 21 146)(11 134 22 145)(23 166 34 155)(24 165 35 176)(25 164 36 175)(26 163 37 174)(27 162 38 173)(28 161 39 172)(29 160 40 171)(30 159 41 170)(31 158 42 169)(32 157 43 168)(33 156 44 167)(45 183 56 194)(46 182 57 193)(47 181 58 192)(48 180 59 191)(49 179 60 190)(50 178 61 189)(51 177 62 188)(52 198 63 187)(53 197 64 186)(54 196 65 185)(55 195 66 184)(67 213 78 202)(68 212 79 201)(69 211 80 200)(70 210 81 199)(71 209 82 220)(72 208 83 219)(73 207 84 218)(74 206 85 217)(75 205 86 216)(76 204 87 215)(77 203 88 214)(89 228 100 239)(90 227 101 238)(91 226 102 237)(92 225 103 236)(93 224 104 235)(94 223 105 234)(95 222 106 233)(96 221 107 232)(97 242 108 231)(98 241 109 230)(99 240 110 229)(111 247 122 258)(112 246 123 257)(113 245 124 256)(114 244 125 255)(115 243 126 254)(116 264 127 253)(117 263 128 252)(118 262 129 251)(119 261 130 250)(120 260 131 249)(121 259 132 248)
G:=sub<Sym(264)| (1,96,62,70,23,115)(2,97,63,71,24,116)(3,98,64,72,25,117)(4,99,65,73,26,118)(5,100,66,74,27,119)(6,101,45,75,28,120)(7,102,46,76,29,121)(8,103,47,77,30,122)(9,104,48,78,31,123)(10,105,49,79,32,124)(11,106,50,80,33,125)(12,107,51,81,34,126)(13,108,52,82,35,127)(14,109,53,83,36,128)(15,110,54,84,37,129)(16,89,55,85,38,130)(17,90,56,86,39,131)(18,91,57,87,40,132)(19,92,58,88,41,111)(20,93,59,67,42,112)(21,94,60,68,43,113)(22,95,61,69,44,114)(133,232,177,199,155,254)(134,233,178,200,156,255)(135,234,179,201,157,256)(136,235,180,202,158,257)(137,236,181,203,159,258)(138,237,182,204,160,259)(139,238,183,205,161,260)(140,239,184,206,162,261)(141,240,185,207,163,262)(142,241,186,208,164,263)(143,242,187,209,165,264)(144,221,188,210,166,243)(145,222,189,211,167,244)(146,223,190,212,168,245)(147,224,191,213,169,246)(148,225,192,214,170,247)(149,226,193,215,171,248)(150,227,194,216,172,249)(151,228,195,217,173,250)(152,229,196,218,174,251)(153,230,197,219,175,252)(154,231,198,220,176,253), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242)(243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264), (1,144,12,133)(2,143,13,154)(3,142,14,153)(4,141,15,152)(5,140,16,151)(6,139,17,150)(7,138,18,149)(8,137,19,148)(9,136,20,147)(10,135,21,146)(11,134,22,145)(23,166,34,155)(24,165,35,176)(25,164,36,175)(26,163,37,174)(27,162,38,173)(28,161,39,172)(29,160,40,171)(30,159,41,170)(31,158,42,169)(32,157,43,168)(33,156,44,167)(45,183,56,194)(46,182,57,193)(47,181,58,192)(48,180,59,191)(49,179,60,190)(50,178,61,189)(51,177,62,188)(52,198,63,187)(53,197,64,186)(54,196,65,185)(55,195,66,184)(67,213,78,202)(68,212,79,201)(69,211,80,200)(70,210,81,199)(71,209,82,220)(72,208,83,219)(73,207,84,218)(74,206,85,217)(75,205,86,216)(76,204,87,215)(77,203,88,214)(89,228,100,239)(90,227,101,238)(91,226,102,237)(92,225,103,236)(93,224,104,235)(94,223,105,234)(95,222,106,233)(96,221,107,232)(97,242,108,231)(98,241,109,230)(99,240,110,229)(111,247,122,258)(112,246,123,257)(113,245,124,256)(114,244,125,255)(115,243,126,254)(116,264,127,253)(117,263,128,252)(118,262,129,251)(119,261,130,250)(120,260,131,249)(121,259,132,248)>;
G:=Group( (1,96,62,70,23,115)(2,97,63,71,24,116)(3,98,64,72,25,117)(4,99,65,73,26,118)(5,100,66,74,27,119)(6,101,45,75,28,120)(7,102,46,76,29,121)(8,103,47,77,30,122)(9,104,48,78,31,123)(10,105,49,79,32,124)(11,106,50,80,33,125)(12,107,51,81,34,126)(13,108,52,82,35,127)(14,109,53,83,36,128)(15,110,54,84,37,129)(16,89,55,85,38,130)(17,90,56,86,39,131)(18,91,57,87,40,132)(19,92,58,88,41,111)(20,93,59,67,42,112)(21,94,60,68,43,113)(22,95,61,69,44,114)(133,232,177,199,155,254)(134,233,178,200,156,255)(135,234,179,201,157,256)(136,235,180,202,158,257)(137,236,181,203,159,258)(138,237,182,204,160,259)(139,238,183,205,161,260)(140,239,184,206,162,261)(141,240,185,207,163,262)(142,241,186,208,164,263)(143,242,187,209,165,264)(144,221,188,210,166,243)(145,222,189,211,167,244)(146,223,190,212,168,245)(147,224,191,213,169,246)(148,225,192,214,170,247)(149,226,193,215,171,248)(150,227,194,216,172,249)(151,228,195,217,173,250)(152,229,196,218,174,251)(153,230,197,219,175,252)(154,231,198,220,176,253), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242)(243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264), (1,144,12,133)(2,143,13,154)(3,142,14,153)(4,141,15,152)(5,140,16,151)(6,139,17,150)(7,138,18,149)(8,137,19,148)(9,136,20,147)(10,135,21,146)(11,134,22,145)(23,166,34,155)(24,165,35,176)(25,164,36,175)(26,163,37,174)(27,162,38,173)(28,161,39,172)(29,160,40,171)(30,159,41,170)(31,158,42,169)(32,157,43,168)(33,156,44,167)(45,183,56,194)(46,182,57,193)(47,181,58,192)(48,180,59,191)(49,179,60,190)(50,178,61,189)(51,177,62,188)(52,198,63,187)(53,197,64,186)(54,196,65,185)(55,195,66,184)(67,213,78,202)(68,212,79,201)(69,211,80,200)(70,210,81,199)(71,209,82,220)(72,208,83,219)(73,207,84,218)(74,206,85,217)(75,205,86,216)(76,204,87,215)(77,203,88,214)(89,228,100,239)(90,227,101,238)(91,226,102,237)(92,225,103,236)(93,224,104,235)(94,223,105,234)(95,222,106,233)(96,221,107,232)(97,242,108,231)(98,241,109,230)(99,240,110,229)(111,247,122,258)(112,246,123,257)(113,245,124,256)(114,244,125,255)(115,243,126,254)(116,264,127,253)(117,263,128,252)(118,262,129,251)(119,261,130,250)(120,260,131,249)(121,259,132,248) );
G=PermutationGroup([[(1,96,62,70,23,115),(2,97,63,71,24,116),(3,98,64,72,25,117),(4,99,65,73,26,118),(5,100,66,74,27,119),(6,101,45,75,28,120),(7,102,46,76,29,121),(8,103,47,77,30,122),(9,104,48,78,31,123),(10,105,49,79,32,124),(11,106,50,80,33,125),(12,107,51,81,34,126),(13,108,52,82,35,127),(14,109,53,83,36,128),(15,110,54,84,37,129),(16,89,55,85,38,130),(17,90,56,86,39,131),(18,91,57,87,40,132),(19,92,58,88,41,111),(20,93,59,67,42,112),(21,94,60,68,43,113),(22,95,61,69,44,114),(133,232,177,199,155,254),(134,233,178,200,156,255),(135,234,179,201,157,256),(136,235,180,202,158,257),(137,236,181,203,159,258),(138,237,182,204,160,259),(139,238,183,205,161,260),(140,239,184,206,162,261),(141,240,185,207,163,262),(142,241,186,208,164,263),(143,242,187,209,165,264),(144,221,188,210,166,243),(145,222,189,211,167,244),(146,223,190,212,168,245),(147,224,191,213,169,246),(148,225,192,214,170,247),(149,226,193,215,171,248),(150,227,194,216,172,249),(151,228,195,217,173,250),(152,229,196,218,174,251),(153,230,197,219,175,252),(154,231,198,220,176,253)], [(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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242),(243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)], [(1,144,12,133),(2,143,13,154),(3,142,14,153),(4,141,15,152),(5,140,16,151),(6,139,17,150),(7,138,18,149),(8,137,19,148),(9,136,20,147),(10,135,21,146),(11,134,22,145),(23,166,34,155),(24,165,35,176),(25,164,36,175),(26,163,37,174),(27,162,38,173),(28,161,39,172),(29,160,40,171),(30,159,41,170),(31,158,42,169),(32,157,43,168),(33,156,44,167),(45,183,56,194),(46,182,57,193),(47,181,58,192),(48,180,59,191),(49,179,60,190),(50,178,61,189),(51,177,62,188),(52,198,63,187),(53,197,64,186),(54,196,65,185),(55,195,66,184),(67,213,78,202),(68,212,79,201),(69,211,80,200),(70,210,81,199),(71,209,82,220),(72,208,83,219),(73,207,84,218),(74,206,85,217),(75,205,86,216),(76,204,87,215),(77,203,88,214),(89,228,100,239),(90,227,101,238),(91,226,102,237),(92,225,103,236),(93,224,104,235),(94,223,105,234),(95,222,106,233),(96,221,107,232),(97,242,108,231),(98,241,109,230),(99,240,110,229),(111,247,122,258),(112,246,123,257),(113,245,124,256),(114,244,125,255),(115,243,126,254),(116,264,127,253),(117,263,128,252),(118,262,129,251),(119,261,130,250),(120,260,131,249),(121,259,132,248)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 11A | ··· | 11E | 12A | ··· | 12H | 22A | ··· | 22O | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 11 | ··· | 11 | 12 | ··· | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D11 | Dic11 | D22 | C3×D11 | C3×Dic11 | C6×D11 |
| kernel | C6×Dic11 | C3×Dic11 | C2×C66 | C2×Dic11 | C66 | Dic11 | C2×C22 | C22 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 5 | 10 | 5 | 10 | 20 | 10 |
Matrix representation of C6×Dic11 ►in GL4(𝔽397) generated by
| 396 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 362 | 0 |
| 0 | 0 | 0 | 362 |
| 1 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 396 | 1 |
| 0 | 0 | 106 | 290 |
| 1 | 0 | 0 | 0 |
| 0 | 334 | 0 | 0 |
| 0 | 0 | 42 | 108 |
| 0 | 0 | 377 | 355 |
G:=sub<GL(4,GF(397))| [396,0,0,0,0,396,0,0,0,0,362,0,0,0,0,362],[1,0,0,0,0,396,0,0,0,0,396,106,0,0,1,290],[1,0,0,0,0,334,0,0,0,0,42,377,0,0,108,355] >;
C6×Dic11 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{11} % in TeX
G:=Group("C6xDic11"); // GroupNames label
G:=SmallGroup(264,16);
// by ID
G=gap.SmallGroup(264,16);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-11,60,6004]);
// Polycyclic
G:=Group<a,b,c|a^6=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export