direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C33, C4.C66, C44.3C6, C12.3C22, C132.7C2, C66.24C22, C22.7(C2×C6), C2.2(C2×C66), C6.7(C2×C22), SmallGroup(264,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C33
G = < a,b,c | a33=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C33
►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)
(1 175 147 91)(2 176 148 92)(3 177 149 93)(4 178 150 94)(5 179 151 95)(6 180 152 96)(7 181 153 97)(8 182 154 98)(9 183 155 99)(10 184 156 67)(11 185 157 68)(12 186 158 69)(13 187 159 70)(14 188 160 71)(15 189 161 72)(16 190 162 73)(17 191 163 74)(18 192 164 75)(19 193 165 76)(20 194 133 77)(21 195 134 78)(22 196 135 79)(23 197 136 80)(24 198 137 81)(25 166 138 82)(26 167 139 83)(27 168 140 84)(28 169 141 85)(29 170 142 86)(30 171 143 87)(31 172 144 88)(32 173 145 89)(33 174 146 90)(34 217 100 250)(35 218 101 251)(36 219 102 252)(37 220 103 253)(38 221 104 254)(39 222 105 255)(40 223 106 256)(41 224 107 257)(42 225 108 258)(43 226 109 259)(44 227 110 260)(45 228 111 261)(46 229 112 262)(47 230 113 263)(48 231 114 264)(49 199 115 232)(50 200 116 233)(51 201 117 234)(52 202 118 235)(53 203 119 236)(54 204 120 237)(55 205 121 238)(56 206 122 239)(57 207 123 240)(58 208 124 241)(59 209 125 242)(60 210 126 243)(61 211 127 244)(62 212 128 245)(63 213 129 246)(64 214 130 247)(65 215 131 248)(66 216 132 249)
(1 35 147 101)(2 36 148 102)(3 37 149 103)(4 38 150 104)(5 39 151 105)(6 40 152 106)(7 41 153 107)(8 42 154 108)(9 43 155 109)(10 44 156 110)(11 45 157 111)(12 46 158 112)(13 47 159 113)(14 48 160 114)(15 49 161 115)(16 50 162 116)(17 51 163 117)(18 52 164 118)(19 53 165 119)(20 54 133 120)(21 55 134 121)(22 56 135 122)(23 57 136 123)(24 58 137 124)(25 59 138 125)(26 60 139 126)(27 61 140 127)(28 62 141 128)(29 63 142 129)(30 64 143 130)(31 65 144 131)(32 66 145 132)(33 34 146 100)(67 227 184 260)(68 228 185 261)(69 229 186 262)(70 230 187 263)(71 231 188 264)(72 199 189 232)(73 200 190 233)(74 201 191 234)(75 202 192 235)(76 203 193 236)(77 204 194 237)(78 205 195 238)(79 206 196 239)(80 207 197 240)(81 208 198 241)(82 209 166 242)(83 210 167 243)(84 211 168 244)(85 212 169 245)(86 213 170 246)(87 214 171 247)(88 215 172 248)(89 216 173 249)(90 217 174 250)(91 218 175 251)(92 219 176 252)(93 220 177 253)(94 221 178 254)(95 222 179 255)(96 223 180 256)(97 224 181 257)(98 225 182 258)(99 226 183 259)
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), (1,175,147,91)(2,176,148,92)(3,177,149,93)(4,178,150,94)(5,179,151,95)(6,180,152,96)(7,181,153,97)(8,182,154,98)(9,183,155,99)(10,184,156,67)(11,185,157,68)(12,186,158,69)(13,187,159,70)(14,188,160,71)(15,189,161,72)(16,190,162,73)(17,191,163,74)(18,192,164,75)(19,193,165,76)(20,194,133,77)(21,195,134,78)(22,196,135,79)(23,197,136,80)(24,198,137,81)(25,166,138,82)(26,167,139,83)(27,168,140,84)(28,169,141,85)(29,170,142,86)(30,171,143,87)(31,172,144,88)(32,173,145,89)(33,174,146,90)(34,217,100,250)(35,218,101,251)(36,219,102,252)(37,220,103,253)(38,221,104,254)(39,222,105,255)(40,223,106,256)(41,224,107,257)(42,225,108,258)(43,226,109,259)(44,227,110,260)(45,228,111,261)(46,229,112,262)(47,230,113,263)(48,231,114,264)(49,199,115,232)(50,200,116,233)(51,201,117,234)(52,202,118,235)(53,203,119,236)(54,204,120,237)(55,205,121,238)(56,206,122,239)(57,207,123,240)(58,208,124,241)(59,209,125,242)(60,210,126,243)(61,211,127,244)(62,212,128,245)(63,213,129,246)(64,214,130,247)(65,215,131,248)(66,216,132,249), (1,35,147,101)(2,36,148,102)(3,37,149,103)(4,38,150,104)(5,39,151,105)(6,40,152,106)(7,41,153,107)(8,42,154,108)(9,43,155,109)(10,44,156,110)(11,45,157,111)(12,46,158,112)(13,47,159,113)(14,48,160,114)(15,49,161,115)(16,50,162,116)(17,51,163,117)(18,52,164,118)(19,53,165,119)(20,54,133,120)(21,55,134,121)(22,56,135,122)(23,57,136,123)(24,58,137,124)(25,59,138,125)(26,60,139,126)(27,61,140,127)(28,62,141,128)(29,63,142,129)(30,64,143,130)(31,65,144,131)(32,66,145,132)(33,34,146,100)(67,227,184,260)(68,228,185,261)(69,229,186,262)(70,230,187,263)(71,231,188,264)(72,199,189,232)(73,200,190,233)(74,201,191,234)(75,202,192,235)(76,203,193,236)(77,204,194,237)(78,205,195,238)(79,206,196,239)(80,207,197,240)(81,208,198,241)(82,209,166,242)(83,210,167,243)(84,211,168,244)(85,212,169,245)(86,213,170,246)(87,214,171,247)(88,215,172,248)(89,216,173,249)(90,217,174,250)(91,218,175,251)(92,219,176,252)(93,220,177,253)(94,221,178,254)(95,222,179,255)(96,223,180,256)(97,224,181,257)(98,225,182,258)(99,226,183,259)>;
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), (1,175,147,91)(2,176,148,92)(3,177,149,93)(4,178,150,94)(5,179,151,95)(6,180,152,96)(7,181,153,97)(8,182,154,98)(9,183,155,99)(10,184,156,67)(11,185,157,68)(12,186,158,69)(13,187,159,70)(14,188,160,71)(15,189,161,72)(16,190,162,73)(17,191,163,74)(18,192,164,75)(19,193,165,76)(20,194,133,77)(21,195,134,78)(22,196,135,79)(23,197,136,80)(24,198,137,81)(25,166,138,82)(26,167,139,83)(27,168,140,84)(28,169,141,85)(29,170,142,86)(30,171,143,87)(31,172,144,88)(32,173,145,89)(33,174,146,90)(34,217,100,250)(35,218,101,251)(36,219,102,252)(37,220,103,253)(38,221,104,254)(39,222,105,255)(40,223,106,256)(41,224,107,257)(42,225,108,258)(43,226,109,259)(44,227,110,260)(45,228,111,261)(46,229,112,262)(47,230,113,263)(48,231,114,264)(49,199,115,232)(50,200,116,233)(51,201,117,234)(52,202,118,235)(53,203,119,236)(54,204,120,237)(55,205,121,238)(56,206,122,239)(57,207,123,240)(58,208,124,241)(59,209,125,242)(60,210,126,243)(61,211,127,244)(62,212,128,245)(63,213,129,246)(64,214,130,247)(65,215,131,248)(66,216,132,249), (1,35,147,101)(2,36,148,102)(3,37,149,103)(4,38,150,104)(5,39,151,105)(6,40,152,106)(7,41,153,107)(8,42,154,108)(9,43,155,109)(10,44,156,110)(11,45,157,111)(12,46,158,112)(13,47,159,113)(14,48,160,114)(15,49,161,115)(16,50,162,116)(17,51,163,117)(18,52,164,118)(19,53,165,119)(20,54,133,120)(21,55,134,121)(22,56,135,122)(23,57,136,123)(24,58,137,124)(25,59,138,125)(26,60,139,126)(27,61,140,127)(28,62,141,128)(29,63,142,129)(30,64,143,130)(31,65,144,131)(32,66,145,132)(33,34,146,100)(67,227,184,260)(68,228,185,261)(69,229,186,262)(70,230,187,263)(71,231,188,264)(72,199,189,232)(73,200,190,233)(74,201,191,234)(75,202,192,235)(76,203,193,236)(77,204,194,237)(78,205,195,238)(79,206,196,239)(80,207,197,240)(81,208,198,241)(82,209,166,242)(83,210,167,243)(84,211,168,244)(85,212,169,245)(86,213,170,246)(87,214,171,247)(88,215,172,248)(89,216,173,249)(90,217,174,250)(91,218,175,251)(92,219,176,252)(93,220,177,253)(94,221,178,254)(95,222,179,255)(96,223,180,256)(97,224,181,257)(98,225,182,258)(99,226,183,259) );
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)], [(1,175,147,91),(2,176,148,92),(3,177,149,93),(4,178,150,94),(5,179,151,95),(6,180,152,96),(7,181,153,97),(8,182,154,98),(9,183,155,99),(10,184,156,67),(11,185,157,68),(12,186,158,69),(13,187,159,70),(14,188,160,71),(15,189,161,72),(16,190,162,73),(17,191,163,74),(18,192,164,75),(19,193,165,76),(20,194,133,77),(21,195,134,78),(22,196,135,79),(23,197,136,80),(24,198,137,81),(25,166,138,82),(26,167,139,83),(27,168,140,84),(28,169,141,85),(29,170,142,86),(30,171,143,87),(31,172,144,88),(32,173,145,89),(33,174,146,90),(34,217,100,250),(35,218,101,251),(36,219,102,252),(37,220,103,253),(38,221,104,254),(39,222,105,255),(40,223,106,256),(41,224,107,257),(42,225,108,258),(43,226,109,259),(44,227,110,260),(45,228,111,261),(46,229,112,262),(47,230,113,263),(48,231,114,264),(49,199,115,232),(50,200,116,233),(51,201,117,234),(52,202,118,235),(53,203,119,236),(54,204,120,237),(55,205,121,238),(56,206,122,239),(57,207,123,240),(58,208,124,241),(59,209,125,242),(60,210,126,243),(61,211,127,244),(62,212,128,245),(63,213,129,246),(64,214,130,247),(65,215,131,248),(66,216,132,249)], [(1,35,147,101),(2,36,148,102),(3,37,149,103),(4,38,150,104),(5,39,151,105),(6,40,152,106),(7,41,153,107),(8,42,154,108),(9,43,155,109),(10,44,156,110),(11,45,157,111),(12,46,158,112),(13,47,159,113),(14,48,160,114),(15,49,161,115),(16,50,162,116),(17,51,163,117),(18,52,164,118),(19,53,165,119),(20,54,133,120),(21,55,134,121),(22,56,135,122),(23,57,136,123),(24,58,137,124),(25,59,138,125),(26,60,139,126),(27,61,140,127),(28,62,141,128),(29,63,142,129),(30,64,143,130),(31,65,144,131),(32,66,145,132),(33,34,146,100),(67,227,184,260),(68,228,185,261),(69,229,186,262),(70,230,187,263),(71,231,188,264),(72,199,189,232),(73,200,190,233),(74,201,191,234),(75,202,192,235),(76,203,193,236),(77,204,194,237),(78,205,195,238),(79,206,196,239),(80,207,197,240),(81,208,198,241),(82,209,166,242),(83,210,167,243),(84,211,168,244),(85,212,169,245),(86,213,170,246),(87,214,171,247),(88,215,172,248),(89,216,173,249),(90,217,174,250),(91,218,175,251),(92,219,176,252),(93,220,177,253),(94,221,178,254),(95,222,179,255),(96,223,180,256),(97,224,181,257),(98,225,182,258),(99,226,183,259)]])
165 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 11A | ··· | 11J | 12A | ··· | 12F | 22A | ··· | 22J | 33A | ··· | 33T | 44A | ··· | 44AD | 66A | ··· | 66T | 132A | ··· | 132BH |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 11 | ··· | 11 | 12 | ··· | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 | Q8 | C3×Q8 | Q8×C11 | Q8×C33 |
| kernel | Q8×C33 | C132 | Q8×C11 | C44 | C3×Q8 | C12 | Q8 | C4 | C33 | C11 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 10 | 30 | 20 | 60 | 1 | 2 | 10 | 20 |
Matrix representation of Q8×C33 ►in GL2(𝔽397) generated by
| 120 | 0 |
| 0 | 120 |
| 1 | 395 |
| 1 | 396 |
| 26 | 210 |
| 131 | 371 |
G:=sub<GL(2,GF(397))| [120,0,0,120],[1,1,395,396],[26,131,210,371] >;
Q8×C33 in GAP, Magma, Sage, TeX
Q_8\times C_{33} % in TeX
G:=Group("Q8xC33"); // GroupNames label
G:=SmallGroup(264,30);
// by ID
G=gap.SmallGroup(264,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-2,660,1341,666]);
// Polycyclic
G:=Group<a,b,c|a^33=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export