metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.3Q8, C4.3Dic18, C12.4Dic6, C4⋊C4.6D9, C18.6(C2×Q8), (C2×C4).44D18, (C2×C12).50D6, C9⋊3(C42.C2), Dic9⋊C4.4C2, C4⋊Dic9.11C2, (C2×C36).9C22, (C4×Dic9).2C2, C6.33(C2×Dic6), C2.8(C2×Dic18), C3.(C4.Dic6), C18.25(C4○D4), (C2×C18).31C23, C2.4(Q8⋊3D9), C6.81(D4⋊2S3), C2.12(D4⋊2D9), C6.37(Q8⋊3S3), (C2×Dic9).8C22, C22.48(C22×D9), (C9×C4⋊C4).7C2, (C3×C4⋊C4).8S3, (C2×C6).188(C22×S3), SmallGroup(288,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.3Q8
G = < a,b,c | a36=b4=1, c2=b2, bab-1=a19, cac-1=a17, cbc-1=a18b-1 >
Subgroups: 292 in 84 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, C9, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C18, C2×Dic3, C2×C12, C2×C12, C42.C2, Dic9, C36, C36, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C4.Dic6, C4×Dic9, Dic9⋊C4, C4⋊Dic9, C4⋊Dic9, C9×C4⋊C4, C36.3Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, D9, Dic6, C22×S3, C42.C2, D18, C2×Dic6, D4⋊2S3, Q8⋊3S3, Dic18, C22×D9, C4.Dic6, C2×Dic18, D4⋊2D9, Q8⋊3D9, C36.3Q8
►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 71 286 111)(2 54 287 130)(3 37 288 113)(4 56 253 132)(5 39 254 115)(6 58 255 134)(7 41 256 117)(8 60 257 136)(9 43 258 119)(10 62 259 138)(11 45 260 121)(12 64 261 140)(13 47 262 123)(14 66 263 142)(15 49 264 125)(16 68 265 144)(17 51 266 127)(18 70 267 110)(19 53 268 129)(20 72 269 112)(21 55 270 131)(22 38 271 114)(23 57 272 133)(24 40 273 116)(25 59 274 135)(26 42 275 118)(27 61 276 137)(28 44 277 120)(29 63 278 139)(30 46 279 122)(31 65 280 141)(32 48 281 124)(33 67 282 143)(34 50 283 126)(35 69 284 109)(36 52 285 128)(73 177 184 244)(74 160 185 227)(75 179 186 246)(76 162 187 229)(77 145 188 248)(78 164 189 231)(79 147 190 250)(80 166 191 233)(81 149 192 252)(82 168 193 235)(83 151 194 218)(84 170 195 237)(85 153 196 220)(86 172 197 239)(87 155 198 222)(88 174 199 241)(89 157 200 224)(90 176 201 243)(91 159 202 226)(92 178 203 245)(93 161 204 228)(94 180 205 247)(95 163 206 230)(96 146 207 249)(97 165 208 232)(98 148 209 251)(99 167 210 234)(100 150 211 217)(101 169 212 236)(102 152 213 219)(103 171 214 238)(104 154 215 221)(105 173 216 240)(106 156 181 223)(107 175 182 242)(108 158 183 225)
(1 88 286 199)(2 105 287 216)(3 86 288 197)(4 103 253 214)(5 84 254 195)(6 101 255 212)(7 82 256 193)(8 99 257 210)(9 80 258 191)(10 97 259 208)(11 78 260 189)(12 95 261 206)(13 76 262 187)(14 93 263 204)(15 74 264 185)(16 91 265 202)(17 108 266 183)(18 89 267 200)(19 106 268 181)(20 87 269 198)(21 104 270 215)(22 85 271 196)(23 102 272 213)(24 83 273 194)(25 100 274 211)(26 81 275 192)(27 98 276 209)(28 79 277 190)(29 96 278 207)(30 77 279 188)(31 94 280 205)(32 75 281 186)(33 92 282 203)(34 73 283 184)(35 90 284 201)(36 107 285 182)(37 221 113 154)(38 238 114 171)(39 219 115 152)(40 236 116 169)(41 217 117 150)(42 234 118 167)(43 251 119 148)(44 232 120 165)(45 249 121 146)(46 230 122 163)(47 247 123 180)(48 228 124 161)(49 245 125 178)(50 226 126 159)(51 243 127 176)(52 224 128 157)(53 241 129 174)(54 222 130 155)(55 239 131 172)(56 220 132 153)(57 237 133 170)(58 218 134 151)(59 235 135 168)(60 252 136 149)(61 233 137 166)(62 250 138 147)(63 231 139 164)(64 248 140 145)(65 229 141 162)(66 246 142 179)(67 227 143 160)(68 244 144 177)(69 225 109 158)(70 242 110 175)(71 223 111 156)(72 240 112 173)
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), (1,71,286,111)(2,54,287,130)(3,37,288,113)(4,56,253,132)(5,39,254,115)(6,58,255,134)(7,41,256,117)(8,60,257,136)(9,43,258,119)(10,62,259,138)(11,45,260,121)(12,64,261,140)(13,47,262,123)(14,66,263,142)(15,49,264,125)(16,68,265,144)(17,51,266,127)(18,70,267,110)(19,53,268,129)(20,72,269,112)(21,55,270,131)(22,38,271,114)(23,57,272,133)(24,40,273,116)(25,59,274,135)(26,42,275,118)(27,61,276,137)(28,44,277,120)(29,63,278,139)(30,46,279,122)(31,65,280,141)(32,48,281,124)(33,67,282,143)(34,50,283,126)(35,69,284,109)(36,52,285,128)(73,177,184,244)(74,160,185,227)(75,179,186,246)(76,162,187,229)(77,145,188,248)(78,164,189,231)(79,147,190,250)(80,166,191,233)(81,149,192,252)(82,168,193,235)(83,151,194,218)(84,170,195,237)(85,153,196,220)(86,172,197,239)(87,155,198,222)(88,174,199,241)(89,157,200,224)(90,176,201,243)(91,159,202,226)(92,178,203,245)(93,161,204,228)(94,180,205,247)(95,163,206,230)(96,146,207,249)(97,165,208,232)(98,148,209,251)(99,167,210,234)(100,150,211,217)(101,169,212,236)(102,152,213,219)(103,171,214,238)(104,154,215,221)(105,173,216,240)(106,156,181,223)(107,175,182,242)(108,158,183,225), (1,88,286,199)(2,105,287,216)(3,86,288,197)(4,103,253,214)(5,84,254,195)(6,101,255,212)(7,82,256,193)(8,99,257,210)(9,80,258,191)(10,97,259,208)(11,78,260,189)(12,95,261,206)(13,76,262,187)(14,93,263,204)(15,74,264,185)(16,91,265,202)(17,108,266,183)(18,89,267,200)(19,106,268,181)(20,87,269,198)(21,104,270,215)(22,85,271,196)(23,102,272,213)(24,83,273,194)(25,100,274,211)(26,81,275,192)(27,98,276,209)(28,79,277,190)(29,96,278,207)(30,77,279,188)(31,94,280,205)(32,75,281,186)(33,92,282,203)(34,73,283,184)(35,90,284,201)(36,107,285,182)(37,221,113,154)(38,238,114,171)(39,219,115,152)(40,236,116,169)(41,217,117,150)(42,234,118,167)(43,251,119,148)(44,232,120,165)(45,249,121,146)(46,230,122,163)(47,247,123,180)(48,228,124,161)(49,245,125,178)(50,226,126,159)(51,243,127,176)(52,224,128,157)(53,241,129,174)(54,222,130,155)(55,239,131,172)(56,220,132,153)(57,237,133,170)(58,218,134,151)(59,235,135,168)(60,252,136,149)(61,233,137,166)(62,250,138,147)(63,231,139,164)(64,248,140,145)(65,229,141,162)(66,246,142,179)(67,227,143,160)(68,244,144,177)(69,225,109,158)(70,242,110,175)(71,223,111,156)(72,240,112,173)>;
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), (1,71,286,111)(2,54,287,130)(3,37,288,113)(4,56,253,132)(5,39,254,115)(6,58,255,134)(7,41,256,117)(8,60,257,136)(9,43,258,119)(10,62,259,138)(11,45,260,121)(12,64,261,140)(13,47,262,123)(14,66,263,142)(15,49,264,125)(16,68,265,144)(17,51,266,127)(18,70,267,110)(19,53,268,129)(20,72,269,112)(21,55,270,131)(22,38,271,114)(23,57,272,133)(24,40,273,116)(25,59,274,135)(26,42,275,118)(27,61,276,137)(28,44,277,120)(29,63,278,139)(30,46,279,122)(31,65,280,141)(32,48,281,124)(33,67,282,143)(34,50,283,126)(35,69,284,109)(36,52,285,128)(73,177,184,244)(74,160,185,227)(75,179,186,246)(76,162,187,229)(77,145,188,248)(78,164,189,231)(79,147,190,250)(80,166,191,233)(81,149,192,252)(82,168,193,235)(83,151,194,218)(84,170,195,237)(85,153,196,220)(86,172,197,239)(87,155,198,222)(88,174,199,241)(89,157,200,224)(90,176,201,243)(91,159,202,226)(92,178,203,245)(93,161,204,228)(94,180,205,247)(95,163,206,230)(96,146,207,249)(97,165,208,232)(98,148,209,251)(99,167,210,234)(100,150,211,217)(101,169,212,236)(102,152,213,219)(103,171,214,238)(104,154,215,221)(105,173,216,240)(106,156,181,223)(107,175,182,242)(108,158,183,225), (1,88,286,199)(2,105,287,216)(3,86,288,197)(4,103,253,214)(5,84,254,195)(6,101,255,212)(7,82,256,193)(8,99,257,210)(9,80,258,191)(10,97,259,208)(11,78,260,189)(12,95,261,206)(13,76,262,187)(14,93,263,204)(15,74,264,185)(16,91,265,202)(17,108,266,183)(18,89,267,200)(19,106,268,181)(20,87,269,198)(21,104,270,215)(22,85,271,196)(23,102,272,213)(24,83,273,194)(25,100,274,211)(26,81,275,192)(27,98,276,209)(28,79,277,190)(29,96,278,207)(30,77,279,188)(31,94,280,205)(32,75,281,186)(33,92,282,203)(34,73,283,184)(35,90,284,201)(36,107,285,182)(37,221,113,154)(38,238,114,171)(39,219,115,152)(40,236,116,169)(41,217,117,150)(42,234,118,167)(43,251,119,148)(44,232,120,165)(45,249,121,146)(46,230,122,163)(47,247,123,180)(48,228,124,161)(49,245,125,178)(50,226,126,159)(51,243,127,176)(52,224,128,157)(53,241,129,174)(54,222,130,155)(55,239,131,172)(56,220,132,153)(57,237,133,170)(58,218,134,151)(59,235,135,168)(60,252,136,149)(61,233,137,166)(62,250,138,147)(63,231,139,164)(64,248,140,145)(65,229,141,162)(66,246,142,179)(67,227,143,160)(68,244,144,177)(69,225,109,158)(70,242,110,175)(71,223,111,156)(72,240,112,173) );
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)], [(1,71,286,111),(2,54,287,130),(3,37,288,113),(4,56,253,132),(5,39,254,115),(6,58,255,134),(7,41,256,117),(8,60,257,136),(9,43,258,119),(10,62,259,138),(11,45,260,121),(12,64,261,140),(13,47,262,123),(14,66,263,142),(15,49,264,125),(16,68,265,144),(17,51,266,127),(18,70,267,110),(19,53,268,129),(20,72,269,112),(21,55,270,131),(22,38,271,114),(23,57,272,133),(24,40,273,116),(25,59,274,135),(26,42,275,118),(27,61,276,137),(28,44,277,120),(29,63,278,139),(30,46,279,122),(31,65,280,141),(32,48,281,124),(33,67,282,143),(34,50,283,126),(35,69,284,109),(36,52,285,128),(73,177,184,244),(74,160,185,227),(75,179,186,246),(76,162,187,229),(77,145,188,248),(78,164,189,231),(79,147,190,250),(80,166,191,233),(81,149,192,252),(82,168,193,235),(83,151,194,218),(84,170,195,237),(85,153,196,220),(86,172,197,239),(87,155,198,222),(88,174,199,241),(89,157,200,224),(90,176,201,243),(91,159,202,226),(92,178,203,245),(93,161,204,228),(94,180,205,247),(95,163,206,230),(96,146,207,249),(97,165,208,232),(98,148,209,251),(99,167,210,234),(100,150,211,217),(101,169,212,236),(102,152,213,219),(103,171,214,238),(104,154,215,221),(105,173,216,240),(106,156,181,223),(107,175,182,242),(108,158,183,225)], [(1,88,286,199),(2,105,287,216),(3,86,288,197),(4,103,253,214),(5,84,254,195),(6,101,255,212),(7,82,256,193),(8,99,257,210),(9,80,258,191),(10,97,259,208),(11,78,260,189),(12,95,261,206),(13,76,262,187),(14,93,263,204),(15,74,264,185),(16,91,265,202),(17,108,266,183),(18,89,267,200),(19,106,268,181),(20,87,269,198),(21,104,270,215),(22,85,271,196),(23,102,272,213),(24,83,273,194),(25,100,274,211),(26,81,275,192),(27,98,276,209),(28,79,277,190),(29,96,278,207),(30,77,279,188),(31,94,280,205),(32,75,281,186),(33,92,282,203),(34,73,283,184),(35,90,284,201),(36,107,285,182),(37,221,113,154),(38,238,114,171),(39,219,115,152),(40,236,116,169),(41,217,117,150),(42,234,118,167),(43,251,119,148),(44,232,120,165),(45,249,121,146),(46,230,122,163),(47,247,123,180),(48,228,124,161),(49,245,125,178),(50,226,126,159),(51,243,127,176),(52,224,128,157),(53,241,129,174),(54,222,130,155),(55,239,131,172),(56,220,132,153),(57,237,133,170),(58,218,134,151),(59,235,135,168),(60,252,136,149),(61,233,137,166),(62,250,138,147),(63,231,139,164),(64,248,140,145),(65,229,141,162),(66,246,142,179),(67,227,143,160),(68,244,144,177),(69,225,109,158),(70,242,110,175),(71,223,111,156),(72,240,112,173)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | - | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | C4○D4 | D9 | Dic6 | D18 | Dic18 | D4⋊2S3 | Q8⋊3S3 | D4⋊2D9 | Q8⋊3D9 |
| kernel | C36.3Q8 | C4×Dic9 | Dic9⋊C4 | C4⋊Dic9 | C9×C4⋊C4 | C3×C4⋊C4 | C36 | C2×C12 | C18 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 2 | 3 | 4 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C36.3Q8 ►in GL4(𝔽37) generated by
| 30 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 11 | 27 |
| 0 | 0 | 27 | 26 |
| 6 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
G:=sub<GL(4,GF(37))| [30,0,0,0,0,21,0,0,0,0,11,27,0,0,27,26],[6,0,0,0,0,31,0,0,0,0,0,36,0,0,1,0],[0,36,0,0,1,0,0,0,0,0,6,0,0,0,0,6] >;
C36.3Q8 in GAP, Magma, Sage, TeX
C_{36}._3Q_8 % in TeX
G:=Group("C36.3Q8"); // GroupNames label
G:=SmallGroup(288,100);
// by ID
G=gap.SmallGroup(288,100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,590,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^18*b^-1>;
// generators/relations