metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.6Q8, C42.5D9, C4.6Dic18, C12.22Dic6, (C4×C36).3C2, (C4×C12).5S3, C18.3(C2×Q8), (C2×C4).63D18, C4⋊Dic9.5C2, C9⋊1(C42.C2), C18.2(C4○D4), (C2×C12).337D6, Dic9⋊C4.1C2, C6.30(C2×Dic6), C2.5(C2×Dic18), C3.(C12.6Q8), C6.72(C4○D12), (C2×C18).11C23, (C2×C36).86C22, C2.6(D36⋊5C2), (C2×Dic9).2C22, C22.35(C22×D9), (C2×C6).168(C22×S3), SmallGroup(288,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.6Q8
G = < a,b,c | a36=b4=1, c2=a18b2, ab=ba, cac-1=a-1, cbc-1=a18b-1 >
Subgroups: 292 in 84 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, C9, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C18, C18, C2×Dic3, C2×C12, C2×C12, C42.C2, Dic9, C36, C36, C2×C18, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic9, C2×C36, C2×C36, C12.6Q8, Dic9⋊C4, C4⋊Dic9, C4×C36, C36.6Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, D9, Dic6, C22×S3, C42.C2, D18, C2×Dic6, C4○D12, Dic18, C22×D9, C12.6Q8, C2×Dic18, D36⋊5C2, C36.6Q8
►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)
(1 251 104 285)(2 252 105 286)(3 217 106 287)(4 218 107 288)(5 219 108 253)(6 220 73 254)(7 221 74 255)(8 222 75 256)(9 223 76 257)(10 224 77 258)(11 225 78 259)(12 226 79 260)(13 227 80 261)(14 228 81 262)(15 229 82 263)(16 230 83 264)(17 231 84 265)(18 232 85 266)(19 233 86 267)(20 234 87 268)(21 235 88 269)(22 236 89 270)(23 237 90 271)(24 238 91 272)(25 239 92 273)(26 240 93 274)(27 241 94 275)(28 242 95 276)(29 243 96 277)(30 244 97 278)(31 245 98 279)(32 246 99 280)(33 247 100 281)(34 248 101 282)(35 249 102 283)(36 250 103 284)(37 215 171 115)(38 216 172 116)(39 181 173 117)(40 182 174 118)(41 183 175 119)(42 184 176 120)(43 185 177 121)(44 186 178 122)(45 187 179 123)(46 188 180 124)(47 189 145 125)(48 190 146 126)(49 191 147 127)(50 192 148 128)(51 193 149 129)(52 194 150 130)(53 195 151 131)(54 196 152 132)(55 197 153 133)(56 198 154 134)(57 199 155 135)(58 200 156 136)(59 201 157 137)(60 202 158 138)(61 203 159 139)(62 204 160 140)(63 205 161 141)(64 206 162 142)(65 207 163 143)(66 208 164 144)(67 209 165 109)(68 210 166 110)(69 211 167 111)(70 212 168 112)(71 213 169 113)(72 214 170 114)
(1 58 86 174)(2 57 87 173)(3 56 88 172)(4 55 89 171)(5 54 90 170)(6 53 91 169)(7 52 92 168)(8 51 93 167)(9 50 94 166)(10 49 95 165)(11 48 96 164)(12 47 97 163)(13 46 98 162)(14 45 99 161)(15 44 100 160)(16 43 101 159)(17 42 102 158)(18 41 103 157)(19 40 104 156)(20 39 105 155)(21 38 106 154)(22 37 107 153)(23 72 108 152)(24 71 73 151)(25 70 74 150)(26 69 75 149)(27 68 76 148)(28 67 77 147)(29 66 78 146)(30 65 79 145)(31 64 80 180)(32 63 81 179)(33 62 82 178)(34 61 83 177)(35 60 84 176)(36 59 85 175)(109 276 191 224)(110 275 192 223)(111 274 193 222)(112 273 194 221)(113 272 195 220)(114 271 196 219)(115 270 197 218)(116 269 198 217)(117 268 199 252)(118 267 200 251)(119 266 201 250)(120 265 202 249)(121 264 203 248)(122 263 204 247)(123 262 205 246)(124 261 206 245)(125 260 207 244)(126 259 208 243)(127 258 209 242)(128 257 210 241)(129 256 211 240)(130 255 212 239)(131 254 213 238)(132 253 214 237)(133 288 215 236)(134 287 216 235)(135 286 181 234)(136 285 182 233)(137 284 183 232)(138 283 184 231)(139 282 185 230)(140 281 186 229)(141 280 187 228)(142 279 188 227)(143 278 189 226)(144 277 190 225)
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), (1,251,104,285)(2,252,105,286)(3,217,106,287)(4,218,107,288)(5,219,108,253)(6,220,73,254)(7,221,74,255)(8,222,75,256)(9,223,76,257)(10,224,77,258)(11,225,78,259)(12,226,79,260)(13,227,80,261)(14,228,81,262)(15,229,82,263)(16,230,83,264)(17,231,84,265)(18,232,85,266)(19,233,86,267)(20,234,87,268)(21,235,88,269)(22,236,89,270)(23,237,90,271)(24,238,91,272)(25,239,92,273)(26,240,93,274)(27,241,94,275)(28,242,95,276)(29,243,96,277)(30,244,97,278)(31,245,98,279)(32,246,99,280)(33,247,100,281)(34,248,101,282)(35,249,102,283)(36,250,103,284)(37,215,171,115)(38,216,172,116)(39,181,173,117)(40,182,174,118)(41,183,175,119)(42,184,176,120)(43,185,177,121)(44,186,178,122)(45,187,179,123)(46,188,180,124)(47,189,145,125)(48,190,146,126)(49,191,147,127)(50,192,148,128)(51,193,149,129)(52,194,150,130)(53,195,151,131)(54,196,152,132)(55,197,153,133)(56,198,154,134)(57,199,155,135)(58,200,156,136)(59,201,157,137)(60,202,158,138)(61,203,159,139)(62,204,160,140)(63,205,161,141)(64,206,162,142)(65,207,163,143)(66,208,164,144)(67,209,165,109)(68,210,166,110)(69,211,167,111)(70,212,168,112)(71,213,169,113)(72,214,170,114), (1,58,86,174)(2,57,87,173)(3,56,88,172)(4,55,89,171)(5,54,90,170)(6,53,91,169)(7,52,92,168)(8,51,93,167)(9,50,94,166)(10,49,95,165)(11,48,96,164)(12,47,97,163)(13,46,98,162)(14,45,99,161)(15,44,100,160)(16,43,101,159)(17,42,102,158)(18,41,103,157)(19,40,104,156)(20,39,105,155)(21,38,106,154)(22,37,107,153)(23,72,108,152)(24,71,73,151)(25,70,74,150)(26,69,75,149)(27,68,76,148)(28,67,77,147)(29,66,78,146)(30,65,79,145)(31,64,80,180)(32,63,81,179)(33,62,82,178)(34,61,83,177)(35,60,84,176)(36,59,85,175)(109,276,191,224)(110,275,192,223)(111,274,193,222)(112,273,194,221)(113,272,195,220)(114,271,196,219)(115,270,197,218)(116,269,198,217)(117,268,199,252)(118,267,200,251)(119,266,201,250)(120,265,202,249)(121,264,203,248)(122,263,204,247)(123,262,205,246)(124,261,206,245)(125,260,207,244)(126,259,208,243)(127,258,209,242)(128,257,210,241)(129,256,211,240)(130,255,212,239)(131,254,213,238)(132,253,214,237)(133,288,215,236)(134,287,216,235)(135,286,181,234)(136,285,182,233)(137,284,183,232)(138,283,184,231)(139,282,185,230)(140,281,186,229)(141,280,187,228)(142,279,188,227)(143,278,189,226)(144,277,190,225)>;
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), (1,251,104,285)(2,252,105,286)(3,217,106,287)(4,218,107,288)(5,219,108,253)(6,220,73,254)(7,221,74,255)(8,222,75,256)(9,223,76,257)(10,224,77,258)(11,225,78,259)(12,226,79,260)(13,227,80,261)(14,228,81,262)(15,229,82,263)(16,230,83,264)(17,231,84,265)(18,232,85,266)(19,233,86,267)(20,234,87,268)(21,235,88,269)(22,236,89,270)(23,237,90,271)(24,238,91,272)(25,239,92,273)(26,240,93,274)(27,241,94,275)(28,242,95,276)(29,243,96,277)(30,244,97,278)(31,245,98,279)(32,246,99,280)(33,247,100,281)(34,248,101,282)(35,249,102,283)(36,250,103,284)(37,215,171,115)(38,216,172,116)(39,181,173,117)(40,182,174,118)(41,183,175,119)(42,184,176,120)(43,185,177,121)(44,186,178,122)(45,187,179,123)(46,188,180,124)(47,189,145,125)(48,190,146,126)(49,191,147,127)(50,192,148,128)(51,193,149,129)(52,194,150,130)(53,195,151,131)(54,196,152,132)(55,197,153,133)(56,198,154,134)(57,199,155,135)(58,200,156,136)(59,201,157,137)(60,202,158,138)(61,203,159,139)(62,204,160,140)(63,205,161,141)(64,206,162,142)(65,207,163,143)(66,208,164,144)(67,209,165,109)(68,210,166,110)(69,211,167,111)(70,212,168,112)(71,213,169,113)(72,214,170,114), (1,58,86,174)(2,57,87,173)(3,56,88,172)(4,55,89,171)(5,54,90,170)(6,53,91,169)(7,52,92,168)(8,51,93,167)(9,50,94,166)(10,49,95,165)(11,48,96,164)(12,47,97,163)(13,46,98,162)(14,45,99,161)(15,44,100,160)(16,43,101,159)(17,42,102,158)(18,41,103,157)(19,40,104,156)(20,39,105,155)(21,38,106,154)(22,37,107,153)(23,72,108,152)(24,71,73,151)(25,70,74,150)(26,69,75,149)(27,68,76,148)(28,67,77,147)(29,66,78,146)(30,65,79,145)(31,64,80,180)(32,63,81,179)(33,62,82,178)(34,61,83,177)(35,60,84,176)(36,59,85,175)(109,276,191,224)(110,275,192,223)(111,274,193,222)(112,273,194,221)(113,272,195,220)(114,271,196,219)(115,270,197,218)(116,269,198,217)(117,268,199,252)(118,267,200,251)(119,266,201,250)(120,265,202,249)(121,264,203,248)(122,263,204,247)(123,262,205,246)(124,261,206,245)(125,260,207,244)(126,259,208,243)(127,258,209,242)(128,257,210,241)(129,256,211,240)(130,255,212,239)(131,254,213,238)(132,253,214,237)(133,288,215,236)(134,287,216,235)(135,286,181,234)(136,285,182,233)(137,284,183,232)(138,283,184,231)(139,282,185,230)(140,281,186,229)(141,280,187,228)(142,279,188,227)(143,278,189,226)(144,277,190,225) );
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)], [(1,251,104,285),(2,252,105,286),(3,217,106,287),(4,218,107,288),(5,219,108,253),(6,220,73,254),(7,221,74,255),(8,222,75,256),(9,223,76,257),(10,224,77,258),(11,225,78,259),(12,226,79,260),(13,227,80,261),(14,228,81,262),(15,229,82,263),(16,230,83,264),(17,231,84,265),(18,232,85,266),(19,233,86,267),(20,234,87,268),(21,235,88,269),(22,236,89,270),(23,237,90,271),(24,238,91,272),(25,239,92,273),(26,240,93,274),(27,241,94,275),(28,242,95,276),(29,243,96,277),(30,244,97,278),(31,245,98,279),(32,246,99,280),(33,247,100,281),(34,248,101,282),(35,249,102,283),(36,250,103,284),(37,215,171,115),(38,216,172,116),(39,181,173,117),(40,182,174,118),(41,183,175,119),(42,184,176,120),(43,185,177,121),(44,186,178,122),(45,187,179,123),(46,188,180,124),(47,189,145,125),(48,190,146,126),(49,191,147,127),(50,192,148,128),(51,193,149,129),(52,194,150,130),(53,195,151,131),(54,196,152,132),(55,197,153,133),(56,198,154,134),(57,199,155,135),(58,200,156,136),(59,201,157,137),(60,202,158,138),(61,203,159,139),(62,204,160,140),(63,205,161,141),(64,206,162,142),(65,207,163,143),(66,208,164,144),(67,209,165,109),(68,210,166,110),(69,211,167,111),(70,212,168,112),(71,213,169,113),(72,214,170,114)], [(1,58,86,174),(2,57,87,173),(3,56,88,172),(4,55,89,171),(5,54,90,170),(6,53,91,169),(7,52,92,168),(8,51,93,167),(9,50,94,166),(10,49,95,165),(11,48,96,164),(12,47,97,163),(13,46,98,162),(14,45,99,161),(15,44,100,160),(16,43,101,159),(17,42,102,158),(18,41,103,157),(19,40,104,156),(20,39,105,155),(21,38,106,154),(22,37,107,153),(23,72,108,152),(24,71,73,151),(25,70,74,150),(26,69,75,149),(27,68,76,148),(28,67,77,147),(29,66,78,146),(30,65,79,145),(31,64,80,180),(32,63,81,179),(33,62,82,178),(34,61,83,177),(35,60,84,176),(36,59,85,175),(109,276,191,224),(110,275,192,223),(111,274,193,222),(112,273,194,221),(113,272,195,220),(114,271,196,219),(115,270,197,218),(116,269,198,217),(117,268,199,252),(118,267,200,251),(119,266,201,250),(120,265,202,249),(121,264,203,248),(122,263,204,247),(123,262,205,246),(124,261,206,245),(125,260,207,244),(126,259,208,243),(127,258,209,242),(128,257,210,241),(129,256,211,240),(130,255,212,239),(131,254,213,238),(132,253,214,237),(133,288,215,236),(134,287,216,235),(135,286,181,234),(136,285,182,233),(137,284,183,232),(138,283,184,231),(139,282,185,230),(140,281,186,229),(141,280,187,228),(142,279,188,227),(143,278,189,226),(144,277,190,225)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | C4○D4 | D9 | Dic6 | D18 | C4○D12 | Dic18 | D36⋊5C2 |
| kernel | C36.6Q8 | Dic9⋊C4 | C4⋊Dic9 | C4×C36 | C4×C12 | C36 | C2×C12 | C18 | C42 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 1 | 2 | 3 | 4 | 3 | 4 | 9 | 8 | 12 | 24 |
Matrix representation of C36.6Q8 ►in GL4(𝔽37) generated by
| 0 | 6 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 17 | 31 |
| 0 | 0 | 6 | 11 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 5 | 27 |
| 0 | 0 | 10 | 32 |
| 21 | 12 | 0 | 0 |
| 25 | 16 | 0 | 0 |
| 0 | 0 | 36 | 29 |
| 0 | 0 | 28 | 1 |
G:=sub<GL(4,GF(37))| [0,6,0,0,6,0,0,0,0,0,17,6,0,0,31,11],[6,0,0,0,0,6,0,0,0,0,5,10,0,0,27,32],[21,25,0,0,12,16,0,0,0,0,36,28,0,0,29,1] >;
C36.6Q8 in GAP, Magma, Sage, TeX
C_{36}._6Q_8 % in TeX
G:=Group("C36.6Q8"); // GroupNames label
G:=SmallGroup(288,80);
// by ID
G=gap.SmallGroup(288,80);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=a^18*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^18*b^-1>;
// generators/relations