metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C33⋊Q8, C22.7D6, C6.7D22, C3⋊1Dic22, C11⋊1Dic6, Dic11.S3, Dic3.D11, C66.7C22, Dic33.2C2, C2.7(S3×D11), (C3×Dic11).1C2, (C11×Dic3).1C2, SmallGroup(264,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊Q8
G = < a,b,c | a33=b4=1, c2=b2, bab-1=a23, cac-1=a10, cbc-1=b-1 >
Smallest permutation representation of C33⋊Q8
►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)
(1 108 62 85)(2 131 63 75)(3 121 64 98)(4 111 65 88)(5 101 66 78)(6 124 34 68)(7 114 35 91)(8 104 36 81)(9 127 37 71)(10 117 38 94)(11 107 39 84)(12 130 40 74)(13 120 41 97)(14 110 42 87)(15 100 43 77)(16 123 44 67)(17 113 45 90)(18 103 46 80)(19 126 47 70)(20 116 48 93)(21 106 49 83)(22 129 50 73)(23 119 51 96)(24 109 52 86)(25 132 53 76)(26 122 54 99)(27 112 55 89)(28 102 56 79)(29 125 57 69)(30 115 58 92)(31 105 59 82)(32 128 60 72)(33 118 61 95)(133 230 189 254)(134 220 190 244)(135 210 191 234)(136 200 192 257)(137 223 193 247)(138 213 194 237)(139 203 195 260)(140 226 196 250)(141 216 197 240)(142 206 198 263)(143 229 166 253)(144 219 167 243)(145 209 168 233)(146 199 169 256)(147 222 170 246)(148 212 171 236)(149 202 172 259)(150 225 173 249)(151 215 174 239)(152 205 175 262)(153 228 176 252)(154 218 177 242)(155 208 178 232)(156 231 179 255)(157 221 180 245)(158 211 181 235)(159 201 182 258)(160 224 183 248)(161 214 184 238)(162 204 185 261)(163 227 186 251)(164 217 187 241)(165 207 188 264)
(1 189 62 133)(2 166 63 143)(3 176 64 153)(4 186 65 163)(5 196 66 140)(6 173 34 150)(7 183 35 160)(8 193 36 137)(9 170 37 147)(10 180 38 157)(11 190 39 134)(12 167 40 144)(13 177 41 154)(14 187 42 164)(15 197 43 141)(16 174 44 151)(17 184 45 161)(18 194 46 138)(19 171 47 148)(20 181 48 158)(21 191 49 135)(22 168 50 145)(23 178 51 155)(24 188 52 165)(25 198 53 142)(26 175 54 152)(27 185 55 162)(28 195 56 139)(29 172 57 149)(30 182 58 159)(31 192 59 136)(32 169 60 146)(33 179 61 156)(67 239 123 215)(68 249 124 225)(69 259 125 202)(70 236 126 212)(71 246 127 222)(72 256 128 199)(73 233 129 209)(74 243 130 219)(75 253 131 229)(76 263 132 206)(77 240 100 216)(78 250 101 226)(79 260 102 203)(80 237 103 213)(81 247 104 223)(82 257 105 200)(83 234 106 210)(84 244 107 220)(85 254 108 230)(86 264 109 207)(87 241 110 217)(88 251 111 227)(89 261 112 204)(90 238 113 214)(91 248 114 224)(92 258 115 201)(93 235 116 211)(94 245 117 221)(95 255 118 231)(96 232 119 208)(97 242 120 218)(98 252 121 228)(99 262 122 205)
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), (1,108,62,85)(2,131,63,75)(3,121,64,98)(4,111,65,88)(5,101,66,78)(6,124,34,68)(7,114,35,91)(8,104,36,81)(9,127,37,71)(10,117,38,94)(11,107,39,84)(12,130,40,74)(13,120,41,97)(14,110,42,87)(15,100,43,77)(16,123,44,67)(17,113,45,90)(18,103,46,80)(19,126,47,70)(20,116,48,93)(21,106,49,83)(22,129,50,73)(23,119,51,96)(24,109,52,86)(25,132,53,76)(26,122,54,99)(27,112,55,89)(28,102,56,79)(29,125,57,69)(30,115,58,92)(31,105,59,82)(32,128,60,72)(33,118,61,95)(133,230,189,254)(134,220,190,244)(135,210,191,234)(136,200,192,257)(137,223,193,247)(138,213,194,237)(139,203,195,260)(140,226,196,250)(141,216,197,240)(142,206,198,263)(143,229,166,253)(144,219,167,243)(145,209,168,233)(146,199,169,256)(147,222,170,246)(148,212,171,236)(149,202,172,259)(150,225,173,249)(151,215,174,239)(152,205,175,262)(153,228,176,252)(154,218,177,242)(155,208,178,232)(156,231,179,255)(157,221,180,245)(158,211,181,235)(159,201,182,258)(160,224,183,248)(161,214,184,238)(162,204,185,261)(163,227,186,251)(164,217,187,241)(165,207,188,264), (1,189,62,133)(2,166,63,143)(3,176,64,153)(4,186,65,163)(5,196,66,140)(6,173,34,150)(7,183,35,160)(8,193,36,137)(9,170,37,147)(10,180,38,157)(11,190,39,134)(12,167,40,144)(13,177,41,154)(14,187,42,164)(15,197,43,141)(16,174,44,151)(17,184,45,161)(18,194,46,138)(19,171,47,148)(20,181,48,158)(21,191,49,135)(22,168,50,145)(23,178,51,155)(24,188,52,165)(25,198,53,142)(26,175,54,152)(27,185,55,162)(28,195,56,139)(29,172,57,149)(30,182,58,159)(31,192,59,136)(32,169,60,146)(33,179,61,156)(67,239,123,215)(68,249,124,225)(69,259,125,202)(70,236,126,212)(71,246,127,222)(72,256,128,199)(73,233,129,209)(74,243,130,219)(75,253,131,229)(76,263,132,206)(77,240,100,216)(78,250,101,226)(79,260,102,203)(80,237,103,213)(81,247,104,223)(82,257,105,200)(83,234,106,210)(84,244,107,220)(85,254,108,230)(86,264,109,207)(87,241,110,217)(88,251,111,227)(89,261,112,204)(90,238,113,214)(91,248,114,224)(92,258,115,201)(93,235,116,211)(94,245,117,221)(95,255,118,231)(96,232,119,208)(97,242,120,218)(98,252,121,228)(99,262,122,205)>;
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), (1,108,62,85)(2,131,63,75)(3,121,64,98)(4,111,65,88)(5,101,66,78)(6,124,34,68)(7,114,35,91)(8,104,36,81)(9,127,37,71)(10,117,38,94)(11,107,39,84)(12,130,40,74)(13,120,41,97)(14,110,42,87)(15,100,43,77)(16,123,44,67)(17,113,45,90)(18,103,46,80)(19,126,47,70)(20,116,48,93)(21,106,49,83)(22,129,50,73)(23,119,51,96)(24,109,52,86)(25,132,53,76)(26,122,54,99)(27,112,55,89)(28,102,56,79)(29,125,57,69)(30,115,58,92)(31,105,59,82)(32,128,60,72)(33,118,61,95)(133,230,189,254)(134,220,190,244)(135,210,191,234)(136,200,192,257)(137,223,193,247)(138,213,194,237)(139,203,195,260)(140,226,196,250)(141,216,197,240)(142,206,198,263)(143,229,166,253)(144,219,167,243)(145,209,168,233)(146,199,169,256)(147,222,170,246)(148,212,171,236)(149,202,172,259)(150,225,173,249)(151,215,174,239)(152,205,175,262)(153,228,176,252)(154,218,177,242)(155,208,178,232)(156,231,179,255)(157,221,180,245)(158,211,181,235)(159,201,182,258)(160,224,183,248)(161,214,184,238)(162,204,185,261)(163,227,186,251)(164,217,187,241)(165,207,188,264), (1,189,62,133)(2,166,63,143)(3,176,64,153)(4,186,65,163)(5,196,66,140)(6,173,34,150)(7,183,35,160)(8,193,36,137)(9,170,37,147)(10,180,38,157)(11,190,39,134)(12,167,40,144)(13,177,41,154)(14,187,42,164)(15,197,43,141)(16,174,44,151)(17,184,45,161)(18,194,46,138)(19,171,47,148)(20,181,48,158)(21,191,49,135)(22,168,50,145)(23,178,51,155)(24,188,52,165)(25,198,53,142)(26,175,54,152)(27,185,55,162)(28,195,56,139)(29,172,57,149)(30,182,58,159)(31,192,59,136)(32,169,60,146)(33,179,61,156)(67,239,123,215)(68,249,124,225)(69,259,125,202)(70,236,126,212)(71,246,127,222)(72,256,128,199)(73,233,129,209)(74,243,130,219)(75,253,131,229)(76,263,132,206)(77,240,100,216)(78,250,101,226)(79,260,102,203)(80,237,103,213)(81,247,104,223)(82,257,105,200)(83,234,106,210)(84,244,107,220)(85,254,108,230)(86,264,109,207)(87,241,110,217)(88,251,111,227)(89,261,112,204)(90,238,113,214)(91,248,114,224)(92,258,115,201)(93,235,116,211)(94,245,117,221)(95,255,118,231)(96,232,119,208)(97,242,120,218)(98,252,121,228)(99,262,122,205) );
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)], [(1,108,62,85),(2,131,63,75),(3,121,64,98),(4,111,65,88),(5,101,66,78),(6,124,34,68),(7,114,35,91),(8,104,36,81),(9,127,37,71),(10,117,38,94),(11,107,39,84),(12,130,40,74),(13,120,41,97),(14,110,42,87),(15,100,43,77),(16,123,44,67),(17,113,45,90),(18,103,46,80),(19,126,47,70),(20,116,48,93),(21,106,49,83),(22,129,50,73),(23,119,51,96),(24,109,52,86),(25,132,53,76),(26,122,54,99),(27,112,55,89),(28,102,56,79),(29,125,57,69),(30,115,58,92),(31,105,59,82),(32,128,60,72),(33,118,61,95),(133,230,189,254),(134,220,190,244),(135,210,191,234),(136,200,192,257),(137,223,193,247),(138,213,194,237),(139,203,195,260),(140,226,196,250),(141,216,197,240),(142,206,198,263),(143,229,166,253),(144,219,167,243),(145,209,168,233),(146,199,169,256),(147,222,170,246),(148,212,171,236),(149,202,172,259),(150,225,173,249),(151,215,174,239),(152,205,175,262),(153,228,176,252),(154,218,177,242),(155,208,178,232),(156,231,179,255),(157,221,180,245),(158,211,181,235),(159,201,182,258),(160,224,183,248),(161,214,184,238),(162,204,185,261),(163,227,186,251),(164,217,187,241),(165,207,188,264)], [(1,189,62,133),(2,166,63,143),(3,176,64,153),(4,186,65,163),(5,196,66,140),(6,173,34,150),(7,183,35,160),(8,193,36,137),(9,170,37,147),(10,180,38,157),(11,190,39,134),(12,167,40,144),(13,177,41,154),(14,187,42,164),(15,197,43,141),(16,174,44,151),(17,184,45,161),(18,194,46,138),(19,171,47,148),(20,181,48,158),(21,191,49,135),(22,168,50,145),(23,178,51,155),(24,188,52,165),(25,198,53,142),(26,175,54,152),(27,185,55,162),(28,195,56,139),(29,172,57,149),(30,182,58,159),(31,192,59,136),(32,169,60,146),(33,179,61,156),(67,239,123,215),(68,249,124,225),(69,259,125,202),(70,236,126,212),(71,246,127,222),(72,256,128,199),(73,233,129,209),(74,243,130,219),(75,253,131,229),(76,263,132,206),(77,240,100,216),(78,250,101,226),(79,260,102,203),(80,237,103,213),(81,247,104,223),(82,257,105,200),(83,234,106,210),(84,244,107,220),(85,254,108,230),(86,264,109,207),(87,241,110,217),(88,251,111,227),(89,261,112,204),(90,238,113,214),(91,248,114,224),(92,258,115,201),(93,235,116,211),(94,245,117,221),(95,255,118,231),(96,232,119,208),(97,242,120,218),(98,252,121,228),(99,262,122,205)]])
39 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33E | 44A | ··· | 44J | 66A | ··· | 66E |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 6 | 22 | 66 | 2 | 2 | ··· | 2 | 22 | 22 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D11 | Dic6 | D22 | Dic22 | S3×D11 | C33⋊Q8 |
| kernel | C33⋊Q8 | C11×Dic3 | C3×Dic11 | Dic33 | Dic11 | C33 | C22 | Dic3 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 5 | 5 |
Matrix representation of C33⋊Q8 ►in GL4(𝔽397) generated by
| 396 | 1 | 0 | 0 |
| 396 | 0 | 0 | 0 |
| 0 | 0 | 185 | 244 |
| 0 | 0 | 77 | 108 |
| 396 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 128 | 43 |
| 0 | 0 | 256 | 269 |
| 396 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 288 | 367 |
| 0 | 0 | 52 | 109 |
G:=sub<GL(4,GF(397))| [396,396,0,0,1,0,0,0,0,0,185,77,0,0,244,108],[396,0,0,0,1,1,0,0,0,0,128,256,0,0,43,269],[396,0,0,0,0,396,0,0,0,0,288,52,0,0,367,109] >;
C33⋊Q8 in GAP, Magma, Sage, TeX
C_{33}\rtimes Q_8 % in TeX
G:=Group("C33:Q8"); // GroupNames label
G:=SmallGroup(264,11);
// by ID
G=gap.SmallGroup(264,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,61,26,168,6004]);
// Polycyclic
G:=Group<a,b,c|a^33=b^4=1,c^2=b^2,b*a*b^-1=a^23,c*a*c^-1=a^10,c*b*c^-1=b^-1>;
// generators/relations
Export