metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊2Q8, C4.4D36, C4⋊2Dic18, C36.27D4, C42.4D9, C12.37D12, C12.21Dic6, C9⋊1(C4⋊Q8), (C4×C12).4S3, (C4×C36).2C2, C2.4(C2×D36), C18.1(C2×D4), C18.2(C2×Q8), (C2×C4).77D18, C6.30(C2×D12), C3.(C12⋊2Q8), C4⋊Dic9.4C2, (C2×C12).367D6, C6.29(C2×Dic6), C2.4(C2×Dic18), (C2×C18).10C23, (C2×C36).85C22, (C2×Dic18).2C2, (C2×Dic9).1C22, C22.34(C22×D9), (C2×C6).167(C22×S3), SmallGroup(288,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊2Q8
G = < a,b,c | a36=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 396 in 102 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C42, C4⋊C4, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C2×C12, C4⋊Q8, Dic9, C36, C2×C18, C4⋊Dic3, C4×C12, C2×Dic6, Dic18, C2×Dic9, C2×C36, C2×C36, C12⋊2Q8, C4⋊Dic9, C4×C36, C2×Dic18, C36⋊2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, D9, Dic6, D12, C22×S3, C4⋊Q8, D18, C2×Dic6, C2×D12, Dic18, D36, C22×D9, C12⋊2Q8, C2×Dic18, C2×D36, C36⋊2Q8
►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)
(1 109 257 181)(2 110 258 182)(3 111 259 183)(4 112 260 184)(5 113 261 185)(6 114 262 186)(7 115 263 187)(8 116 264 188)(9 117 265 189)(10 118 266 190)(11 119 267 191)(12 120 268 192)(13 121 269 193)(14 122 270 194)(15 123 271 195)(16 124 272 196)(17 125 273 197)(18 126 274 198)(19 127 275 199)(20 128 276 200)(21 129 277 201)(22 130 278 202)(23 131 279 203)(24 132 280 204)(25 133 281 205)(26 134 282 206)(27 135 283 207)(28 136 284 208)(29 137 285 209)(30 138 286 210)(31 139 287 211)(32 140 288 212)(33 141 253 213)(34 142 254 214)(35 143 255 215)(36 144 256 216)(37 92 240 152)(38 93 241 153)(39 94 242 154)(40 95 243 155)(41 96 244 156)(42 97 245 157)(43 98 246 158)(44 99 247 159)(45 100 248 160)(46 101 249 161)(47 102 250 162)(48 103 251 163)(49 104 252 164)(50 105 217 165)(51 106 218 166)(52 107 219 167)(53 108 220 168)(54 73 221 169)(55 74 222 170)(56 75 223 171)(57 76 224 172)(58 77 225 173)(59 78 226 174)(60 79 227 175)(61 80 228 176)(62 81 229 177)(63 82 230 178)(64 83 231 179)(65 84 232 180)(66 85 233 145)(67 86 234 146)(68 87 235 147)(69 88 236 148)(70 89 237 149)(71 90 238 150)(72 91 239 151)
(1 226 257 59)(2 225 258 58)(3 224 259 57)(4 223 260 56)(5 222 261 55)(6 221 262 54)(7 220 263 53)(8 219 264 52)(9 218 265 51)(10 217 266 50)(11 252 267 49)(12 251 268 48)(13 250 269 47)(14 249 270 46)(15 248 271 45)(16 247 272 44)(17 246 273 43)(18 245 274 42)(19 244 275 41)(20 243 276 40)(21 242 277 39)(22 241 278 38)(23 240 279 37)(24 239 280 72)(25 238 281 71)(26 237 282 70)(27 236 283 69)(28 235 284 68)(29 234 285 67)(30 233 286 66)(31 232 287 65)(32 231 288 64)(33 230 253 63)(34 229 254 62)(35 228 255 61)(36 227 256 60)(73 186 169 114)(74 185 170 113)(75 184 171 112)(76 183 172 111)(77 182 173 110)(78 181 174 109)(79 216 175 144)(80 215 176 143)(81 214 177 142)(82 213 178 141)(83 212 179 140)(84 211 180 139)(85 210 145 138)(86 209 146 137)(87 208 147 136)(88 207 148 135)(89 206 149 134)(90 205 150 133)(91 204 151 132)(92 203 152 131)(93 202 153 130)(94 201 154 129)(95 200 155 128)(96 199 156 127)(97 198 157 126)(98 197 158 125)(99 196 159 124)(100 195 160 123)(101 194 161 122)(102 193 162 121)(103 192 163 120)(104 191 164 119)(105 190 165 118)(106 189 166 117)(107 188 167 116)(108 187 168 115)
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), (1,109,257,181)(2,110,258,182)(3,111,259,183)(4,112,260,184)(5,113,261,185)(6,114,262,186)(7,115,263,187)(8,116,264,188)(9,117,265,189)(10,118,266,190)(11,119,267,191)(12,120,268,192)(13,121,269,193)(14,122,270,194)(15,123,271,195)(16,124,272,196)(17,125,273,197)(18,126,274,198)(19,127,275,199)(20,128,276,200)(21,129,277,201)(22,130,278,202)(23,131,279,203)(24,132,280,204)(25,133,281,205)(26,134,282,206)(27,135,283,207)(28,136,284,208)(29,137,285,209)(30,138,286,210)(31,139,287,211)(32,140,288,212)(33,141,253,213)(34,142,254,214)(35,143,255,215)(36,144,256,216)(37,92,240,152)(38,93,241,153)(39,94,242,154)(40,95,243,155)(41,96,244,156)(42,97,245,157)(43,98,246,158)(44,99,247,159)(45,100,248,160)(46,101,249,161)(47,102,250,162)(48,103,251,163)(49,104,252,164)(50,105,217,165)(51,106,218,166)(52,107,219,167)(53,108,220,168)(54,73,221,169)(55,74,222,170)(56,75,223,171)(57,76,224,172)(58,77,225,173)(59,78,226,174)(60,79,227,175)(61,80,228,176)(62,81,229,177)(63,82,230,178)(64,83,231,179)(65,84,232,180)(66,85,233,145)(67,86,234,146)(68,87,235,147)(69,88,236,148)(70,89,237,149)(71,90,238,150)(72,91,239,151), (1,226,257,59)(2,225,258,58)(3,224,259,57)(4,223,260,56)(5,222,261,55)(6,221,262,54)(7,220,263,53)(8,219,264,52)(9,218,265,51)(10,217,266,50)(11,252,267,49)(12,251,268,48)(13,250,269,47)(14,249,270,46)(15,248,271,45)(16,247,272,44)(17,246,273,43)(18,245,274,42)(19,244,275,41)(20,243,276,40)(21,242,277,39)(22,241,278,38)(23,240,279,37)(24,239,280,72)(25,238,281,71)(26,237,282,70)(27,236,283,69)(28,235,284,68)(29,234,285,67)(30,233,286,66)(31,232,287,65)(32,231,288,64)(33,230,253,63)(34,229,254,62)(35,228,255,61)(36,227,256,60)(73,186,169,114)(74,185,170,113)(75,184,171,112)(76,183,172,111)(77,182,173,110)(78,181,174,109)(79,216,175,144)(80,215,176,143)(81,214,177,142)(82,213,178,141)(83,212,179,140)(84,211,180,139)(85,210,145,138)(86,209,146,137)(87,208,147,136)(88,207,148,135)(89,206,149,134)(90,205,150,133)(91,204,151,132)(92,203,152,131)(93,202,153,130)(94,201,154,129)(95,200,155,128)(96,199,156,127)(97,198,157,126)(98,197,158,125)(99,196,159,124)(100,195,160,123)(101,194,161,122)(102,193,162,121)(103,192,163,120)(104,191,164,119)(105,190,165,118)(106,189,166,117)(107,188,167,116)(108,187,168,115)>;
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), (1,109,257,181)(2,110,258,182)(3,111,259,183)(4,112,260,184)(5,113,261,185)(6,114,262,186)(7,115,263,187)(8,116,264,188)(9,117,265,189)(10,118,266,190)(11,119,267,191)(12,120,268,192)(13,121,269,193)(14,122,270,194)(15,123,271,195)(16,124,272,196)(17,125,273,197)(18,126,274,198)(19,127,275,199)(20,128,276,200)(21,129,277,201)(22,130,278,202)(23,131,279,203)(24,132,280,204)(25,133,281,205)(26,134,282,206)(27,135,283,207)(28,136,284,208)(29,137,285,209)(30,138,286,210)(31,139,287,211)(32,140,288,212)(33,141,253,213)(34,142,254,214)(35,143,255,215)(36,144,256,216)(37,92,240,152)(38,93,241,153)(39,94,242,154)(40,95,243,155)(41,96,244,156)(42,97,245,157)(43,98,246,158)(44,99,247,159)(45,100,248,160)(46,101,249,161)(47,102,250,162)(48,103,251,163)(49,104,252,164)(50,105,217,165)(51,106,218,166)(52,107,219,167)(53,108,220,168)(54,73,221,169)(55,74,222,170)(56,75,223,171)(57,76,224,172)(58,77,225,173)(59,78,226,174)(60,79,227,175)(61,80,228,176)(62,81,229,177)(63,82,230,178)(64,83,231,179)(65,84,232,180)(66,85,233,145)(67,86,234,146)(68,87,235,147)(69,88,236,148)(70,89,237,149)(71,90,238,150)(72,91,239,151), (1,226,257,59)(2,225,258,58)(3,224,259,57)(4,223,260,56)(5,222,261,55)(6,221,262,54)(7,220,263,53)(8,219,264,52)(9,218,265,51)(10,217,266,50)(11,252,267,49)(12,251,268,48)(13,250,269,47)(14,249,270,46)(15,248,271,45)(16,247,272,44)(17,246,273,43)(18,245,274,42)(19,244,275,41)(20,243,276,40)(21,242,277,39)(22,241,278,38)(23,240,279,37)(24,239,280,72)(25,238,281,71)(26,237,282,70)(27,236,283,69)(28,235,284,68)(29,234,285,67)(30,233,286,66)(31,232,287,65)(32,231,288,64)(33,230,253,63)(34,229,254,62)(35,228,255,61)(36,227,256,60)(73,186,169,114)(74,185,170,113)(75,184,171,112)(76,183,172,111)(77,182,173,110)(78,181,174,109)(79,216,175,144)(80,215,176,143)(81,214,177,142)(82,213,178,141)(83,212,179,140)(84,211,180,139)(85,210,145,138)(86,209,146,137)(87,208,147,136)(88,207,148,135)(89,206,149,134)(90,205,150,133)(91,204,151,132)(92,203,152,131)(93,202,153,130)(94,201,154,129)(95,200,155,128)(96,199,156,127)(97,198,157,126)(98,197,158,125)(99,196,159,124)(100,195,160,123)(101,194,161,122)(102,193,162,121)(103,192,163,120)(104,191,164,119)(105,190,165,118)(106,189,166,117)(107,188,167,116)(108,187,168,115) );
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)], [(1,109,257,181),(2,110,258,182),(3,111,259,183),(4,112,260,184),(5,113,261,185),(6,114,262,186),(7,115,263,187),(8,116,264,188),(9,117,265,189),(10,118,266,190),(11,119,267,191),(12,120,268,192),(13,121,269,193),(14,122,270,194),(15,123,271,195),(16,124,272,196),(17,125,273,197),(18,126,274,198),(19,127,275,199),(20,128,276,200),(21,129,277,201),(22,130,278,202),(23,131,279,203),(24,132,280,204),(25,133,281,205),(26,134,282,206),(27,135,283,207),(28,136,284,208),(29,137,285,209),(30,138,286,210),(31,139,287,211),(32,140,288,212),(33,141,253,213),(34,142,254,214),(35,143,255,215),(36,144,256,216),(37,92,240,152),(38,93,241,153),(39,94,242,154),(40,95,243,155),(41,96,244,156),(42,97,245,157),(43,98,246,158),(44,99,247,159),(45,100,248,160),(46,101,249,161),(47,102,250,162),(48,103,251,163),(49,104,252,164),(50,105,217,165),(51,106,218,166),(52,107,219,167),(53,108,220,168),(54,73,221,169),(55,74,222,170),(56,75,223,171),(57,76,224,172),(58,77,225,173),(59,78,226,174),(60,79,227,175),(61,80,228,176),(62,81,229,177),(63,82,230,178),(64,83,231,179),(65,84,232,180),(66,85,233,145),(67,86,234,146),(68,87,235,147),(69,88,236,148),(70,89,237,149),(71,90,238,150),(72,91,239,151)], [(1,226,257,59),(2,225,258,58),(3,224,259,57),(4,223,260,56),(5,222,261,55),(6,221,262,54),(7,220,263,53),(8,219,264,52),(9,218,265,51),(10,217,266,50),(11,252,267,49),(12,251,268,48),(13,250,269,47),(14,249,270,46),(15,248,271,45),(16,247,272,44),(17,246,273,43),(18,245,274,42),(19,244,275,41),(20,243,276,40),(21,242,277,39),(22,241,278,38),(23,240,279,37),(24,239,280,72),(25,238,281,71),(26,237,282,70),(27,236,283,69),(28,235,284,68),(29,234,285,67),(30,233,286,66),(31,232,287,65),(32,231,288,64),(33,230,253,63),(34,229,254,62),(35,228,255,61),(36,227,256,60),(73,186,169,114),(74,185,170,113),(75,184,171,112),(76,183,172,111),(77,182,173,110),(78,181,174,109),(79,216,175,144),(80,215,176,143),(81,214,177,142),(82,213,178,141),(83,212,179,140),(84,211,180,139),(85,210,145,138),(86,209,146,137),(87,208,147,136),(88,207,148,135),(89,206,149,134),(90,205,150,133),(91,204,151,132),(92,203,152,131),(93,202,153,130),(94,201,154,129),(95,200,155,128),(96,199,156,127),(97,198,157,126),(98,197,158,125),(99,196,159,124),(100,195,160,123),(101,194,161,122),(102,193,162,121),(103,192,163,120),(104,191,164,119),(105,190,165,118),(106,189,166,117),(107,188,167,116),(108,187,168,115)]])
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 | D4 | Q8 | D6 | D9 | Dic6 | D12 | D18 | Dic18 | D36 |
| kernel | C36⋊2Q8 | C4⋊Dic9 | C4×C36 | C2×Dic18 | C4×C12 | C36 | C36 | C2×C12 | C42 | C12 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 3 | 3 | 8 | 4 | 9 | 24 | 12 |
Matrix representation of C36⋊2Q8 ►in GL4(𝔽37) generated by
| 11 | 31 | 0 | 0 |
| 6 | 17 | 0 | 0 |
| 0 | 0 | 4 | 29 |
| 0 | 0 | 8 | 12 |
| 5 | 10 | 0 | 0 |
| 27 | 32 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 27 | 32 |
| 28 | 36 | 0 | 0 |
| 8 | 9 | 0 | 0 |
| 0 | 0 | 35 | 11 |
| 0 | 0 | 13 | 2 |
G:=sub<GL(4,GF(37))| [11,6,0,0,31,17,0,0,0,0,4,8,0,0,29,12],[5,27,0,0,10,32,0,0,0,0,5,27,0,0,10,32],[28,8,0,0,36,9,0,0,0,0,35,13,0,0,11,2] >;
C36⋊2Q8 in GAP, Magma, Sage, TeX
C_{36}\rtimes_2Q_8 % in TeX
G:=Group("C36:2Q8"); // GroupNames label
G:=SmallGroup(288,79);
// by ID
G=gap.SmallGroup(288,79);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations