metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C18.6D8, C36.1Q8, C18.3Q16, C12.1Dic6, C4.1Dic18, C9⋊C8⋊1C4, C4⋊C4.1D9, C9⋊1(C2.D8), C12.1(C4×S3), C36.1(C2×C4), C4.11(C4×D9), C18.2(C4⋊C4), C2.1(D4⋊D9), (C2×C18).29D4, (C2×C4).35D18, (C2×C12).37D6, C4⋊Dic9.7C2, C3.(C6.Q16), C6.13(D4⋊S3), C6.6(C3⋊Q16), C2.1(C9⋊Q16), C2.3(Dic9⋊C4), (C2×C36).15C22, C6.10(Dic3⋊C4), C22.12(C9⋊D4), (C2×C9⋊C8).1C2, (C9×C4⋊C4).1C2, (C3×C4⋊C4).1S3, (C2×C6).67(C3⋊D4), SmallGroup(288,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.Q8
G = < a,b,c | a36=b4=1, c2=a9b2, bab-1=a19, cac-1=a17, cbc-1=a9b-1 >
Smallest permutation representation of C36.Q8
►Regular action on 288 points
Generators in S288
(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)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)
(1 263 103 211)(2 282 104 194)(3 265 105 213)(4 284 106 196)(5 267 107 215)(6 286 108 198)(7 269 73 181)(8 288 74 200)(9 271 75 183)(10 254 76 202)(11 273 77 185)(12 256 78 204)(13 275 79 187)(14 258 80 206)(15 277 81 189)(16 260 82 208)(17 279 83 191)(18 262 84 210)(19 281 85 193)(20 264 86 212)(21 283 87 195)(22 266 88 214)(23 285 89 197)(24 268 90 216)(25 287 91 199)(26 270 92 182)(27 253 93 201)(28 272 94 184)(29 255 95 203)(30 274 96 186)(31 257 97 205)(32 276 98 188)(33 259 99 207)(34 278 100 190)(35 261 101 209)(36 280 102 192)(37 129 147 240)(38 112 148 223)(39 131 149 242)(40 114 150 225)(41 133 151 244)(42 116 152 227)(43 135 153 246)(44 118 154 229)(45 137 155 248)(46 120 156 231)(47 139 157 250)(48 122 158 233)(49 141 159 252)(50 124 160 235)(51 143 161 218)(52 126 162 237)(53 109 163 220)(54 128 164 239)(55 111 165 222)(56 130 166 241)(57 113 167 224)(58 132 168 243)(59 115 169 226)(60 134 170 245)(61 117 171 228)(62 136 172 247)(63 119 173 230)(64 138 174 249)(65 121 175 232)(66 140 176 251)(67 123 177 234)(68 142 178 217)(69 125 179 236)(70 144 180 219)(71 127 145 238)(72 110 146 221)
(1 62 76 145 19 44 94 163)(2 43 77 162 20 61 95 180)(3 60 78 179 21 42 96 161)(4 41 79 160 22 59 97 178)(5 58 80 177 23 40 98 159)(6 39 81 158 24 57 99 176)(7 56 82 175 25 38 100 157)(8 37 83 156 26 55 101 174)(9 54 84 173 27 72 102 155)(10 71 85 154 28 53 103 172)(11 52 86 171 29 70 104 153)(12 69 87 152 30 51 105 170)(13 50 88 169 31 68 106 151)(14 67 89 150 32 49 107 168)(15 48 90 167 33 66 108 149)(16 65 91 148 34 47 73 166)(17 46 92 165 35 64 74 147)(18 63 93 146 36 45 75 164)(109 272 229 193 127 254 247 211)(110 253 230 210 128 271 248 192)(111 270 231 191 129 288 249 209)(112 287 232 208 130 269 250 190)(113 268 233 189 131 286 251 207)(114 285 234 206 132 267 252 188)(115 266 235 187 133 284 217 205)(116 283 236 204 134 265 218 186)(117 264 237 185 135 282 219 203)(118 281 238 202 136 263 220 184)(119 262 239 183 137 280 221 201)(120 279 240 200 138 261 222 182)(121 260 241 181 139 278 223 199)(122 277 242 198 140 259 224 216)(123 258 243 215 141 276 225 197)(124 275 244 196 142 257 226 214)(125 256 245 213 143 274 227 195)(126 273 246 194 144 255 228 212)
G:=sub<Sym(288)| (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)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,263,103,211)(2,282,104,194)(3,265,105,213)(4,284,106,196)(5,267,107,215)(6,286,108,198)(7,269,73,181)(8,288,74,200)(9,271,75,183)(10,254,76,202)(11,273,77,185)(12,256,78,204)(13,275,79,187)(14,258,80,206)(15,277,81,189)(16,260,82,208)(17,279,83,191)(18,262,84,210)(19,281,85,193)(20,264,86,212)(21,283,87,195)(22,266,88,214)(23,285,89,197)(24,268,90,216)(25,287,91,199)(26,270,92,182)(27,253,93,201)(28,272,94,184)(29,255,95,203)(30,274,96,186)(31,257,97,205)(32,276,98,188)(33,259,99,207)(34,278,100,190)(35,261,101,209)(36,280,102,192)(37,129,147,240)(38,112,148,223)(39,131,149,242)(40,114,150,225)(41,133,151,244)(42,116,152,227)(43,135,153,246)(44,118,154,229)(45,137,155,248)(46,120,156,231)(47,139,157,250)(48,122,158,233)(49,141,159,252)(50,124,160,235)(51,143,161,218)(52,126,162,237)(53,109,163,220)(54,128,164,239)(55,111,165,222)(56,130,166,241)(57,113,167,224)(58,132,168,243)(59,115,169,226)(60,134,170,245)(61,117,171,228)(62,136,172,247)(63,119,173,230)(64,138,174,249)(65,121,175,232)(66,140,176,251)(67,123,177,234)(68,142,178,217)(69,125,179,236)(70,144,180,219)(71,127,145,238)(72,110,146,221), (1,62,76,145,19,44,94,163)(2,43,77,162,20,61,95,180)(3,60,78,179,21,42,96,161)(4,41,79,160,22,59,97,178)(5,58,80,177,23,40,98,159)(6,39,81,158,24,57,99,176)(7,56,82,175,25,38,100,157)(8,37,83,156,26,55,101,174)(9,54,84,173,27,72,102,155)(10,71,85,154,28,53,103,172)(11,52,86,171,29,70,104,153)(12,69,87,152,30,51,105,170)(13,50,88,169,31,68,106,151)(14,67,89,150,32,49,107,168)(15,48,90,167,33,66,108,149)(16,65,91,148,34,47,73,166)(17,46,92,165,35,64,74,147)(18,63,93,146,36,45,75,164)(109,272,229,193,127,254,247,211)(110,253,230,210,128,271,248,192)(111,270,231,191,129,288,249,209)(112,287,232,208,130,269,250,190)(113,268,233,189,131,286,251,207)(114,285,234,206,132,267,252,188)(115,266,235,187,133,284,217,205)(116,283,236,204,134,265,218,186)(117,264,237,185,135,282,219,203)(118,281,238,202,136,263,220,184)(119,262,239,183,137,280,221,201)(120,279,240,200,138,261,222,182)(121,260,241,181,139,278,223,199)(122,277,242,198,140,259,224,216)(123,258,243,215,141,276,225,197)(124,275,244,196,142,257,226,214)(125,256,245,213,143,274,227,195)(126,273,246,194,144,255,228,212)>;
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)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,263,103,211)(2,282,104,194)(3,265,105,213)(4,284,106,196)(5,267,107,215)(6,286,108,198)(7,269,73,181)(8,288,74,200)(9,271,75,183)(10,254,76,202)(11,273,77,185)(12,256,78,204)(13,275,79,187)(14,258,80,206)(15,277,81,189)(16,260,82,208)(17,279,83,191)(18,262,84,210)(19,281,85,193)(20,264,86,212)(21,283,87,195)(22,266,88,214)(23,285,89,197)(24,268,90,216)(25,287,91,199)(26,270,92,182)(27,253,93,201)(28,272,94,184)(29,255,95,203)(30,274,96,186)(31,257,97,205)(32,276,98,188)(33,259,99,207)(34,278,100,190)(35,261,101,209)(36,280,102,192)(37,129,147,240)(38,112,148,223)(39,131,149,242)(40,114,150,225)(41,133,151,244)(42,116,152,227)(43,135,153,246)(44,118,154,229)(45,137,155,248)(46,120,156,231)(47,139,157,250)(48,122,158,233)(49,141,159,252)(50,124,160,235)(51,143,161,218)(52,126,162,237)(53,109,163,220)(54,128,164,239)(55,111,165,222)(56,130,166,241)(57,113,167,224)(58,132,168,243)(59,115,169,226)(60,134,170,245)(61,117,171,228)(62,136,172,247)(63,119,173,230)(64,138,174,249)(65,121,175,232)(66,140,176,251)(67,123,177,234)(68,142,178,217)(69,125,179,236)(70,144,180,219)(71,127,145,238)(72,110,146,221), (1,62,76,145,19,44,94,163)(2,43,77,162,20,61,95,180)(3,60,78,179,21,42,96,161)(4,41,79,160,22,59,97,178)(5,58,80,177,23,40,98,159)(6,39,81,158,24,57,99,176)(7,56,82,175,25,38,100,157)(8,37,83,156,26,55,101,174)(9,54,84,173,27,72,102,155)(10,71,85,154,28,53,103,172)(11,52,86,171,29,70,104,153)(12,69,87,152,30,51,105,170)(13,50,88,169,31,68,106,151)(14,67,89,150,32,49,107,168)(15,48,90,167,33,66,108,149)(16,65,91,148,34,47,73,166)(17,46,92,165,35,64,74,147)(18,63,93,146,36,45,75,164)(109,272,229,193,127,254,247,211)(110,253,230,210,128,271,248,192)(111,270,231,191,129,288,249,209)(112,287,232,208,130,269,250,190)(113,268,233,189,131,286,251,207)(114,285,234,206,132,267,252,188)(115,266,235,187,133,284,217,205)(116,283,236,204,134,265,218,186)(117,264,237,185,135,282,219,203)(118,281,238,202,136,263,220,184)(119,262,239,183,137,280,221,201)(120,279,240,200,138,261,222,182)(121,260,241,181,139,278,223,199)(122,277,242,198,140,259,224,216)(123,258,243,215,141,276,225,197)(124,275,244,196,142,257,226,214)(125,256,245,213,143,274,227,195)(126,273,246,194,144,255,228,212) );
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),(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)], [(1,263,103,211),(2,282,104,194),(3,265,105,213),(4,284,106,196),(5,267,107,215),(6,286,108,198),(7,269,73,181),(8,288,74,200),(9,271,75,183),(10,254,76,202),(11,273,77,185),(12,256,78,204),(13,275,79,187),(14,258,80,206),(15,277,81,189),(16,260,82,208),(17,279,83,191),(18,262,84,210),(19,281,85,193),(20,264,86,212),(21,283,87,195),(22,266,88,214),(23,285,89,197),(24,268,90,216),(25,287,91,199),(26,270,92,182),(27,253,93,201),(28,272,94,184),(29,255,95,203),(30,274,96,186),(31,257,97,205),(32,276,98,188),(33,259,99,207),(34,278,100,190),(35,261,101,209),(36,280,102,192),(37,129,147,240),(38,112,148,223),(39,131,149,242),(40,114,150,225),(41,133,151,244),(42,116,152,227),(43,135,153,246),(44,118,154,229),(45,137,155,248),(46,120,156,231),(47,139,157,250),(48,122,158,233),(49,141,159,252),(50,124,160,235),(51,143,161,218),(52,126,162,237),(53,109,163,220),(54,128,164,239),(55,111,165,222),(56,130,166,241),(57,113,167,224),(58,132,168,243),(59,115,169,226),(60,134,170,245),(61,117,171,228),(62,136,172,247),(63,119,173,230),(64,138,174,249),(65,121,175,232),(66,140,176,251),(67,123,177,234),(68,142,178,217),(69,125,179,236),(70,144,180,219),(71,127,145,238),(72,110,146,221)], [(1,62,76,145,19,44,94,163),(2,43,77,162,20,61,95,180),(3,60,78,179,21,42,96,161),(4,41,79,160,22,59,97,178),(5,58,80,177,23,40,98,159),(6,39,81,158,24,57,99,176),(7,56,82,175,25,38,100,157),(8,37,83,156,26,55,101,174),(9,54,84,173,27,72,102,155),(10,71,85,154,28,53,103,172),(11,52,86,171,29,70,104,153),(12,69,87,152,30,51,105,170),(13,50,88,169,31,68,106,151),(14,67,89,150,32,49,107,168),(15,48,90,167,33,66,108,149),(16,65,91,148,34,47,73,166),(17,46,92,165,35,64,74,147),(18,63,93,146,36,45,75,164),(109,272,229,193,127,254,247,211),(110,253,230,210,128,271,248,192),(111,270,231,191,129,288,249,209),(112,287,232,208,130,269,250,190),(113,268,233,189,131,286,251,207),(114,285,234,206,132,267,252,188),(115,266,235,187,133,284,217,205),(116,283,236,204,134,265,218,186),(117,264,237,185,135,282,219,203),(118,281,238,202,136,263,220,184),(119,262,239,183,137,280,221,201),(120,279,240,200,138,261,222,182),(121,260,241,181,139,278,223,199),(122,277,242,198,140,259,224,216),(123,258,243,215,141,276,225,197),(124,275,244,196,142,257,226,214),(125,256,245,213,143,274,227,195),(126,273,246,194,144,255,228,212)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | + | - | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | D8 | Q16 | D9 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 | D4⋊S3 | C3⋊Q16 | D4⋊D9 | C9⋊Q16 |
| kernel | C36.Q8 | C2×C9⋊C8 | C4⋊Dic9 | C9×C4⋊C4 | C9⋊C8 | C3×C4⋊C4 | C36 | C2×C18 | C2×C12 | C18 | C18 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C36.Q8 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 70 | 0 | 0 |
| 0 | 0 | 3 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 59 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 41 |
| 0 | 0 | 0 | 0 | 16 | 41 |
| 55 | 2 | 0 | 0 | 0 | 0 |
| 20 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 42 | 0 | 0 |
| 0 | 0 | 45 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 41 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,28,3,0,0,0,0,70,31,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[7,59,0,0,0,0,14,66,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,32,16,0,0,0,0,41,41],[55,20,0,0,0,0,2,18,0,0,0,0,0,0,70,45,0,0,0,0,42,3,0,0,0,0,0,0,32,16,0,0,0,0,41,0] >;
C36.Q8 in GAP, Magma, Sage, TeX
C_{36}.Q_8 % in TeX
G:=Group("C36.Q8"); // GroupNames label
G:=SmallGroup(288,14);
// by ID
G=gap.SmallGroup(288,14);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=a^9*b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^9*b^-1>;
// generators/relations
Export