direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×C3⋊C8, C3⋊C88, C33⋊3C8, C6.C44, C66.3C4, C44.4S3, C132.6C2, C12.2C22, C22.2Dic3, C4.2(S3×C11), C2.(C11×Dic3), SmallGroup(264,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C11×C3⋊C8 |
Generators and relations for C11×C3⋊C8
G = < a,b,c | a11=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×C3⋊C8
►Regular action on 264 points
Generators in S264
(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)
(1 166 225)(2 167 226)(3 168 227)(4 169 228)(5 170 229)(6 171 230)(7 172 231)(8 173 221)(9 174 222)(10 175 223)(11 176 224)(12 58 71)(13 59 72)(14 60 73)(15 61 74)(16 62 75)(17 63 76)(18 64 77)(19 65 67)(20 66 68)(21 56 69)(22 57 70)(23 96 82)(24 97 83)(25 98 84)(26 99 85)(27 89 86)(28 90 87)(29 91 88)(30 92 78)(31 93 79)(32 94 80)(33 95 81)(34 107 48)(35 108 49)(36 109 50)(37 110 51)(38 100 52)(39 101 53)(40 102 54)(41 103 55)(42 104 45)(43 105 46)(44 106 47)(111 243 256)(112 244 257)(113 245 258)(114 246 259)(115 247 260)(116 248 261)(117 249 262)(118 250 263)(119 251 264)(120 252 254)(121 253 255)(122 239 179)(123 240 180)(124 241 181)(125 242 182)(126 232 183)(127 233 184)(128 234 185)(129 235 186)(130 236 187)(131 237 177)(132 238 178)(133 206 193)(134 207 194)(135 208 195)(136 209 196)(137 199 197)(138 200 198)(139 201 188)(140 202 189)(141 203 190)(142 204 191)(143 205 192)(144 158 217)(145 159 218)(146 160 219)(147 161 220)(148 162 210)(149 163 211)(150 164 212)(151 165 213)(152 155 214)(153 156 215)(154 157 216)
(1 254 122 93 163 17 208 48)(2 255 123 94 164 18 209 49)(3 256 124 95 165 19 199 50)(4 257 125 96 155 20 200 51)(5 258 126 97 156 21 201 52)(6 259 127 98 157 22 202 53)(7 260 128 99 158 12 203 54)(8 261 129 89 159 13 204 55)(9 262 130 90 160 14 205 45)(10 263 131 91 161 15 206 46)(11 264 132 92 162 16 207 47)(23 214 68 198 110 169 244 242)(24 215 69 188 100 170 245 232)(25 216 70 189 101 171 246 233)(26 217 71 190 102 172 247 234)(27 218 72 191 103 173 248 235)(28 219 73 192 104 174 249 236)(29 220 74 193 105 175 250 237)(30 210 75 194 106 176 251 238)(31 211 76 195 107 166 252 239)(32 212 77 196 108 167 253 240)(33 213 67 197 109 168 243 241)(34 225 120 179 79 149 63 135)(35 226 121 180 80 150 64 136)(36 227 111 181 81 151 65 137)(37 228 112 182 82 152 66 138)(38 229 113 183 83 153 56 139)(39 230 114 184 84 154 57 140)(40 231 115 185 85 144 58 141)(41 221 116 186 86 145 59 142)(42 222 117 187 87 146 60 143)(43 223 118 177 88 147 61 133)(44 224 119 178 78 148 62 134)
G:=sub<Sym(264)| (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), (1,166,225)(2,167,226)(3,168,227)(4,169,228)(5,170,229)(6,171,230)(7,172,231)(8,173,221)(9,174,222)(10,175,223)(11,176,224)(12,58,71)(13,59,72)(14,60,73)(15,61,74)(16,62,75)(17,63,76)(18,64,77)(19,65,67)(20,66,68)(21,56,69)(22,57,70)(23,96,82)(24,97,83)(25,98,84)(26,99,85)(27,89,86)(28,90,87)(29,91,88)(30,92,78)(31,93,79)(32,94,80)(33,95,81)(34,107,48)(35,108,49)(36,109,50)(37,110,51)(38,100,52)(39,101,53)(40,102,54)(41,103,55)(42,104,45)(43,105,46)(44,106,47)(111,243,256)(112,244,257)(113,245,258)(114,246,259)(115,247,260)(116,248,261)(117,249,262)(118,250,263)(119,251,264)(120,252,254)(121,253,255)(122,239,179)(123,240,180)(124,241,181)(125,242,182)(126,232,183)(127,233,184)(128,234,185)(129,235,186)(130,236,187)(131,237,177)(132,238,178)(133,206,193)(134,207,194)(135,208,195)(136,209,196)(137,199,197)(138,200,198)(139,201,188)(140,202,189)(141,203,190)(142,204,191)(143,205,192)(144,158,217)(145,159,218)(146,160,219)(147,161,220)(148,162,210)(149,163,211)(150,164,212)(151,165,213)(152,155,214)(153,156,215)(154,157,216), (1,254,122,93,163,17,208,48)(2,255,123,94,164,18,209,49)(3,256,124,95,165,19,199,50)(4,257,125,96,155,20,200,51)(5,258,126,97,156,21,201,52)(6,259,127,98,157,22,202,53)(7,260,128,99,158,12,203,54)(8,261,129,89,159,13,204,55)(9,262,130,90,160,14,205,45)(10,263,131,91,161,15,206,46)(11,264,132,92,162,16,207,47)(23,214,68,198,110,169,244,242)(24,215,69,188,100,170,245,232)(25,216,70,189,101,171,246,233)(26,217,71,190,102,172,247,234)(27,218,72,191,103,173,248,235)(28,219,73,192,104,174,249,236)(29,220,74,193,105,175,250,237)(30,210,75,194,106,176,251,238)(31,211,76,195,107,166,252,239)(32,212,77,196,108,167,253,240)(33,213,67,197,109,168,243,241)(34,225,120,179,79,149,63,135)(35,226,121,180,80,150,64,136)(36,227,111,181,81,151,65,137)(37,228,112,182,82,152,66,138)(38,229,113,183,83,153,56,139)(39,230,114,184,84,154,57,140)(40,231,115,185,85,144,58,141)(41,221,116,186,86,145,59,142)(42,222,117,187,87,146,60,143)(43,223,118,177,88,147,61,133)(44,224,119,178,78,148,62,134)>;
G:=Group( (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), (1,166,225)(2,167,226)(3,168,227)(4,169,228)(5,170,229)(6,171,230)(7,172,231)(8,173,221)(9,174,222)(10,175,223)(11,176,224)(12,58,71)(13,59,72)(14,60,73)(15,61,74)(16,62,75)(17,63,76)(18,64,77)(19,65,67)(20,66,68)(21,56,69)(22,57,70)(23,96,82)(24,97,83)(25,98,84)(26,99,85)(27,89,86)(28,90,87)(29,91,88)(30,92,78)(31,93,79)(32,94,80)(33,95,81)(34,107,48)(35,108,49)(36,109,50)(37,110,51)(38,100,52)(39,101,53)(40,102,54)(41,103,55)(42,104,45)(43,105,46)(44,106,47)(111,243,256)(112,244,257)(113,245,258)(114,246,259)(115,247,260)(116,248,261)(117,249,262)(118,250,263)(119,251,264)(120,252,254)(121,253,255)(122,239,179)(123,240,180)(124,241,181)(125,242,182)(126,232,183)(127,233,184)(128,234,185)(129,235,186)(130,236,187)(131,237,177)(132,238,178)(133,206,193)(134,207,194)(135,208,195)(136,209,196)(137,199,197)(138,200,198)(139,201,188)(140,202,189)(141,203,190)(142,204,191)(143,205,192)(144,158,217)(145,159,218)(146,160,219)(147,161,220)(148,162,210)(149,163,211)(150,164,212)(151,165,213)(152,155,214)(153,156,215)(154,157,216), (1,254,122,93,163,17,208,48)(2,255,123,94,164,18,209,49)(3,256,124,95,165,19,199,50)(4,257,125,96,155,20,200,51)(5,258,126,97,156,21,201,52)(6,259,127,98,157,22,202,53)(7,260,128,99,158,12,203,54)(8,261,129,89,159,13,204,55)(9,262,130,90,160,14,205,45)(10,263,131,91,161,15,206,46)(11,264,132,92,162,16,207,47)(23,214,68,198,110,169,244,242)(24,215,69,188,100,170,245,232)(25,216,70,189,101,171,246,233)(26,217,71,190,102,172,247,234)(27,218,72,191,103,173,248,235)(28,219,73,192,104,174,249,236)(29,220,74,193,105,175,250,237)(30,210,75,194,106,176,251,238)(31,211,76,195,107,166,252,239)(32,212,77,196,108,167,253,240)(33,213,67,197,109,168,243,241)(34,225,120,179,79,149,63,135)(35,226,121,180,80,150,64,136)(36,227,111,181,81,151,65,137)(37,228,112,182,82,152,66,138)(38,229,113,183,83,153,56,139)(39,230,114,184,84,154,57,140)(40,231,115,185,85,144,58,141)(41,221,116,186,86,145,59,142)(42,222,117,187,87,146,60,143)(43,223,118,177,88,147,61,133)(44,224,119,178,78,148,62,134) );
G=PermutationGroup([[(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)], [(1,166,225),(2,167,226),(3,168,227),(4,169,228),(5,170,229),(6,171,230),(7,172,231),(8,173,221),(9,174,222),(10,175,223),(11,176,224),(12,58,71),(13,59,72),(14,60,73),(15,61,74),(16,62,75),(17,63,76),(18,64,77),(19,65,67),(20,66,68),(21,56,69),(22,57,70),(23,96,82),(24,97,83),(25,98,84),(26,99,85),(27,89,86),(28,90,87),(29,91,88),(30,92,78),(31,93,79),(32,94,80),(33,95,81),(34,107,48),(35,108,49),(36,109,50),(37,110,51),(38,100,52),(39,101,53),(40,102,54),(41,103,55),(42,104,45),(43,105,46),(44,106,47),(111,243,256),(112,244,257),(113,245,258),(114,246,259),(115,247,260),(116,248,261),(117,249,262),(118,250,263),(119,251,264),(120,252,254),(121,253,255),(122,239,179),(123,240,180),(124,241,181),(125,242,182),(126,232,183),(127,233,184),(128,234,185),(129,235,186),(130,236,187),(131,237,177),(132,238,178),(133,206,193),(134,207,194),(135,208,195),(136,209,196),(137,199,197),(138,200,198),(139,201,188),(140,202,189),(141,203,190),(142,204,191),(143,205,192),(144,158,217),(145,159,218),(146,160,219),(147,161,220),(148,162,210),(149,163,211),(150,164,212),(151,165,213),(152,155,214),(153,156,215),(154,157,216)], [(1,254,122,93,163,17,208,48),(2,255,123,94,164,18,209,49),(3,256,124,95,165,19,199,50),(4,257,125,96,155,20,200,51),(5,258,126,97,156,21,201,52),(6,259,127,98,157,22,202,53),(7,260,128,99,158,12,203,54),(8,261,129,89,159,13,204,55),(9,262,130,90,160,14,205,45),(10,263,131,91,161,15,206,46),(11,264,132,92,162,16,207,47),(23,214,68,198,110,169,244,242),(24,215,69,188,100,170,245,232),(25,216,70,189,101,171,246,233),(26,217,71,190,102,172,247,234),(27,218,72,191,103,173,248,235),(28,219,73,192,104,174,249,236),(29,220,74,193,105,175,250,237),(30,210,75,194,106,176,251,238),(31,211,76,195,107,166,252,239),(32,212,77,196,108,167,253,240),(33,213,67,197,109,168,243,241),(34,225,120,179,79,149,63,135),(35,226,121,180,80,150,64,136),(36,227,111,181,81,151,65,137),(37,228,112,182,82,152,66,138),(38,229,113,183,83,153,56,139),(39,230,114,184,84,154,57,140),(40,231,115,185,85,144,58,141),(41,221,116,186,86,145,59,142),(42,222,117,187,87,146,60,143),(43,223,118,177,88,147,61,133),(44,224,119,178,78,148,62,134)]])
132 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 11A | ··· | 11J | 12A | 12B | 22A | ··· | 22J | 33A | ··· | 33J | 44A | ··· | 44T | 66A | ··· | 66J | 88A | ··· | 88AN | 132A | ··· | 132T |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 88 | ··· | 88 | 132 | ··· | 132 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C8 | C11 | C22 | C44 | C88 | S3 | Dic3 | C3⋊C8 | S3×C11 | C11×Dic3 | C11×C3⋊C8 |
| kernel | C11×C3⋊C8 | C132 | C66 | C33 | C3⋊C8 | C12 | C6 | C3 | C44 | C22 | C11 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 40 | 1 | 1 | 2 | 10 | 10 | 20 |
Matrix representation of C11×C3⋊C8 ►in GL3(𝔽1321) generated by
| 1 | 0 | 0 |
| 0 | 205 | 0 |
| 0 | 0 | 205 |
| 1 | 0 | 0 |
| 0 | 1320 | 1320 |
| 0 | 1 | 0 |
| 235 | 0 | 0 |
| 0 | 1137 | 177 |
| 0 | 361 | 184 |
G:=sub<GL(3,GF(1321))| [1,0,0,0,205,0,0,0,205],[1,0,0,0,1320,1,0,1320,0],[235,0,0,0,1137,361,0,177,184] >;
C11×C3⋊C8 in GAP, Magma, Sage, TeX
C_{11}\times C_3\rtimes C_8 % in TeX
G:=Group("C11xC3:C8"); // GroupNames label
G:=SmallGroup(264,1);
// by ID
G=gap.SmallGroup(264,1);
# by ID
G:=PCGroup([5,-2,-11,-2,-2,-3,110,42,4404]);
// Polycyclic
G:=Group<a,b,c|a^11=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export