metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C33⋊1C8, C66.1C4, C44.2S3, C4.2D33, C22.Dic3, C2.Dic33, C6.Dic11, C132.2C2, C12.2D11, C11⋊(C3⋊C8), C3⋊(C11⋊C8), SmallGroup(264,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C33⋊C8 |
Generators and relations for C33⋊C8
G = < a,b | a33=b8=1, bab-1=a-1 >
Smallest permutation representation of C33⋊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)
(1 257 124 172 34 224 74 141)(2 256 125 171 35 223 75 140)(3 255 126 170 36 222 76 139)(4 254 127 169 37 221 77 138)(5 253 128 168 38 220 78 137)(6 252 129 167 39 219 79 136)(7 251 130 166 40 218 80 135)(8 250 131 198 41 217 81 134)(9 249 132 197 42 216 82 133)(10 248 100 196 43 215 83 165)(11 247 101 195 44 214 84 164)(12 246 102 194 45 213 85 163)(13 245 103 193 46 212 86 162)(14 244 104 192 47 211 87 161)(15 243 105 191 48 210 88 160)(16 242 106 190 49 209 89 159)(17 241 107 189 50 208 90 158)(18 240 108 188 51 207 91 157)(19 239 109 187 52 206 92 156)(20 238 110 186 53 205 93 155)(21 237 111 185 54 204 94 154)(22 236 112 184 55 203 95 153)(23 235 113 183 56 202 96 152)(24 234 114 182 57 201 97 151)(25 233 115 181 58 200 98 150)(26 232 116 180 59 199 99 149)(27 264 117 179 60 231 67 148)(28 263 118 178 61 230 68 147)(29 262 119 177 62 229 69 146)(30 261 120 176 63 228 70 145)(31 260 121 175 64 227 71 144)(32 259 122 174 65 226 72 143)(33 258 123 173 66 225 73 142)
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), (1,257,124,172,34,224,74,141)(2,256,125,171,35,223,75,140)(3,255,126,170,36,222,76,139)(4,254,127,169,37,221,77,138)(5,253,128,168,38,220,78,137)(6,252,129,167,39,219,79,136)(7,251,130,166,40,218,80,135)(8,250,131,198,41,217,81,134)(9,249,132,197,42,216,82,133)(10,248,100,196,43,215,83,165)(11,247,101,195,44,214,84,164)(12,246,102,194,45,213,85,163)(13,245,103,193,46,212,86,162)(14,244,104,192,47,211,87,161)(15,243,105,191,48,210,88,160)(16,242,106,190,49,209,89,159)(17,241,107,189,50,208,90,158)(18,240,108,188,51,207,91,157)(19,239,109,187,52,206,92,156)(20,238,110,186,53,205,93,155)(21,237,111,185,54,204,94,154)(22,236,112,184,55,203,95,153)(23,235,113,183,56,202,96,152)(24,234,114,182,57,201,97,151)(25,233,115,181,58,200,98,150)(26,232,116,180,59,199,99,149)(27,264,117,179,60,231,67,148)(28,263,118,178,61,230,68,147)(29,262,119,177,62,229,69,146)(30,261,120,176,63,228,70,145)(31,260,121,175,64,227,71,144)(32,259,122,174,65,226,72,143)(33,258,123,173,66,225,73,142)>;
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), (1,257,124,172,34,224,74,141)(2,256,125,171,35,223,75,140)(3,255,126,170,36,222,76,139)(4,254,127,169,37,221,77,138)(5,253,128,168,38,220,78,137)(6,252,129,167,39,219,79,136)(7,251,130,166,40,218,80,135)(8,250,131,198,41,217,81,134)(9,249,132,197,42,216,82,133)(10,248,100,196,43,215,83,165)(11,247,101,195,44,214,84,164)(12,246,102,194,45,213,85,163)(13,245,103,193,46,212,86,162)(14,244,104,192,47,211,87,161)(15,243,105,191,48,210,88,160)(16,242,106,190,49,209,89,159)(17,241,107,189,50,208,90,158)(18,240,108,188,51,207,91,157)(19,239,109,187,52,206,92,156)(20,238,110,186,53,205,93,155)(21,237,111,185,54,204,94,154)(22,236,112,184,55,203,95,153)(23,235,113,183,56,202,96,152)(24,234,114,182,57,201,97,151)(25,233,115,181,58,200,98,150)(26,232,116,180,59,199,99,149)(27,264,117,179,60,231,67,148)(28,263,118,178,61,230,68,147)(29,262,119,177,62,229,69,146)(30,261,120,176,63,228,70,145)(31,260,121,175,64,227,71,144)(32,259,122,174,65,226,72,143)(33,258,123,173,66,225,73,142) );
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)], [(1,257,124,172,34,224,74,141),(2,256,125,171,35,223,75,140),(3,255,126,170,36,222,76,139),(4,254,127,169,37,221,77,138),(5,253,128,168,38,220,78,137),(6,252,129,167,39,219,79,136),(7,251,130,166,40,218,80,135),(8,250,131,198,41,217,81,134),(9,249,132,197,42,216,82,133),(10,248,100,196,43,215,83,165),(11,247,101,195,44,214,84,164),(12,246,102,194,45,213,85,163),(13,245,103,193,46,212,86,162),(14,244,104,192,47,211,87,161),(15,243,105,191,48,210,88,160),(16,242,106,190,49,209,89,159),(17,241,107,189,50,208,90,158),(18,240,108,188,51,207,91,157),(19,239,109,187,52,206,92,156),(20,238,110,186,53,205,93,155),(21,237,111,185,54,204,94,154),(22,236,112,184,55,203,95,153),(23,235,113,183,56,202,96,152),(24,234,114,182,57,201,97,151),(25,233,115,181,58,200,98,150),(26,232,116,180,59,199,99,149),(27,264,117,179,60,231,67,148),(28,263,118,178,61,230,68,147),(29,262,119,177,62,229,69,146),(30,261,120,176,63,228,70,145),(31,260,121,175,64,227,71,144),(32,259,122,174,65,226,72,143),(33,258,123,173,66,225,73,142)]])
72 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 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 | 132 | ··· | 132 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 33 | 33 | 33 | 33 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | D11 | C3⋊C8 | Dic11 | D33 | C11⋊C8 | Dic33 | C33⋊C8 |
| kernel | C33⋊C8 | C132 | C66 | C33 | C44 | C22 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C33⋊C8 ►in GL2(𝔽1321) generated by
| 847 | 1079 |
| 877 | 863 |
| 315 | 300 |
| 586 | 1006 |
G:=sub<GL(2,GF(1321))| [847,877,1079,863],[315,586,300,1006] >;
C33⋊C8 in GAP, Magma, Sage, TeX
C_{33}\rtimes C_8 % in TeX
G:=Group("C33:C8"); // GroupNames label
G:=SmallGroup(264,3);
// by ID
G=gap.SmallGroup(264,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,10,26,323,6004]);
// Polycyclic
G:=Group<a,b|a^33=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export