metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊Q8, Dic9⋊1Q8, C4⋊1Dic18, Dic9.2D4, C12.3Dic6, C9⋊2(C4⋊Q8), C4⋊C4.4D9, C3.(C12⋊Q8), C2.4(Q8×D9), (C2×C4).8D18, C2.11(D4×D9), (C2×C12).6D6, C6.83(S3×D4), C18.5(C2×Q8), C6.33(S3×Q8), C18.22(C2×D4), Dic9⋊C4.2C2, C4⋊Dic9.10C2, (C2×C36).7C22, (C4×Dic9).1C2, C2.7(C2×Dic18), C6.32(C2×Dic6), (C2×C18).29C23, (C2×Dic18).3C2, C22.46(C22×D9), (C2×Dic9).29C22, (C9×C4⋊C4).5C2, (C3×C4⋊C4).6S3, (C2×C6).186(C22×S3), SmallGroup(288,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊Q8
G = < a,b,c | a36=b4=1, c2=b2, bab-1=a19, cac-1=a17, cbc-1=b-1 >
Subgroups: 396 in 102 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, C9, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C18, Dic6, C2×Dic3, C2×C12, C2×C12, C4⋊Q8, Dic9, Dic9, C36, C36, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, Dic18, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C12⋊Q8, C4×Dic9, Dic9⋊C4, C4⋊Dic9, C9×C4⋊C4, C2×Dic18, C36⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, D9, Dic6, C22×S3, C4⋊Q8, D18, C2×Dic6, S3×D4, S3×Q8, Dic18, C22×D9, C12⋊Q8, C2×Dic18, D4×D9, Q8×D9, 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)
(1 102 251 137)(2 85 252 120)(3 104 217 139)(4 87 218 122)(5 106 219 141)(6 89 220 124)(7 108 221 143)(8 91 222 126)(9 74 223 109)(10 93 224 128)(11 76 225 111)(12 95 226 130)(13 78 227 113)(14 97 228 132)(15 80 229 115)(16 99 230 134)(17 82 231 117)(18 101 232 136)(19 84 233 119)(20 103 234 138)(21 86 235 121)(22 105 236 140)(23 88 237 123)(24 107 238 142)(25 90 239 125)(26 73 240 144)(27 92 241 127)(28 75 242 110)(29 94 243 129)(30 77 244 112)(31 96 245 131)(32 79 246 114)(33 98 247 133)(34 81 248 116)(35 100 249 135)(36 83 250 118)(37 287 153 202)(38 270 154 185)(39 253 155 204)(40 272 156 187)(41 255 157 206)(42 274 158 189)(43 257 159 208)(44 276 160 191)(45 259 161 210)(46 278 162 193)(47 261 163 212)(48 280 164 195)(49 263 165 214)(50 282 166 197)(51 265 167 216)(52 284 168 199)(53 267 169 182)(54 286 170 201)(55 269 171 184)(56 288 172 203)(57 271 173 186)(58 254 174 205)(59 273 175 188)(60 256 176 207)(61 275 177 190)(62 258 178 209)(63 277 179 192)(64 260 180 211)(65 279 145 194)(66 262 146 213)(67 281 147 196)(68 264 148 215)(69 283 149 198)(70 266 150 181)(71 285 151 200)(72 268 152 183)
(1 181 251 266)(2 198 252 283)(3 215 217 264)(4 196 218 281)(5 213 219 262)(6 194 220 279)(7 211 221 260)(8 192 222 277)(9 209 223 258)(10 190 224 275)(11 207 225 256)(12 188 226 273)(13 205 227 254)(14 186 228 271)(15 203 229 288)(16 184 230 269)(17 201 231 286)(18 182 232 267)(19 199 233 284)(20 216 234 265)(21 197 235 282)(22 214 236 263)(23 195 237 280)(24 212 238 261)(25 193 239 278)(26 210 240 259)(27 191 241 276)(28 208 242 257)(29 189 243 274)(30 206 244 255)(31 187 245 272)(32 204 246 253)(33 185 247 270)(34 202 248 287)(35 183 249 268)(36 200 250 285)(37 81 153 116)(38 98 154 133)(39 79 155 114)(40 96 156 131)(41 77 157 112)(42 94 158 129)(43 75 159 110)(44 92 160 127)(45 73 161 144)(46 90 162 125)(47 107 163 142)(48 88 164 123)(49 105 165 140)(50 86 166 121)(51 103 167 138)(52 84 168 119)(53 101 169 136)(54 82 170 117)(55 99 171 134)(56 80 172 115)(57 97 173 132)(58 78 174 113)(59 95 175 130)(60 76 176 111)(61 93 177 128)(62 74 178 109)(63 91 179 126)(64 108 180 143)(65 89 145 124)(66 106 146 141)(67 87 147 122)(68 104 148 139)(69 85 149 120)(70 102 150 137)(71 83 151 118)(72 100 152 135)
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), (1,102,251,137)(2,85,252,120)(3,104,217,139)(4,87,218,122)(5,106,219,141)(6,89,220,124)(7,108,221,143)(8,91,222,126)(9,74,223,109)(10,93,224,128)(11,76,225,111)(12,95,226,130)(13,78,227,113)(14,97,228,132)(15,80,229,115)(16,99,230,134)(17,82,231,117)(18,101,232,136)(19,84,233,119)(20,103,234,138)(21,86,235,121)(22,105,236,140)(23,88,237,123)(24,107,238,142)(25,90,239,125)(26,73,240,144)(27,92,241,127)(28,75,242,110)(29,94,243,129)(30,77,244,112)(31,96,245,131)(32,79,246,114)(33,98,247,133)(34,81,248,116)(35,100,249,135)(36,83,250,118)(37,287,153,202)(38,270,154,185)(39,253,155,204)(40,272,156,187)(41,255,157,206)(42,274,158,189)(43,257,159,208)(44,276,160,191)(45,259,161,210)(46,278,162,193)(47,261,163,212)(48,280,164,195)(49,263,165,214)(50,282,166,197)(51,265,167,216)(52,284,168,199)(53,267,169,182)(54,286,170,201)(55,269,171,184)(56,288,172,203)(57,271,173,186)(58,254,174,205)(59,273,175,188)(60,256,176,207)(61,275,177,190)(62,258,178,209)(63,277,179,192)(64,260,180,211)(65,279,145,194)(66,262,146,213)(67,281,147,196)(68,264,148,215)(69,283,149,198)(70,266,150,181)(71,285,151,200)(72,268,152,183), (1,181,251,266)(2,198,252,283)(3,215,217,264)(4,196,218,281)(5,213,219,262)(6,194,220,279)(7,211,221,260)(8,192,222,277)(9,209,223,258)(10,190,224,275)(11,207,225,256)(12,188,226,273)(13,205,227,254)(14,186,228,271)(15,203,229,288)(16,184,230,269)(17,201,231,286)(18,182,232,267)(19,199,233,284)(20,216,234,265)(21,197,235,282)(22,214,236,263)(23,195,237,280)(24,212,238,261)(25,193,239,278)(26,210,240,259)(27,191,241,276)(28,208,242,257)(29,189,243,274)(30,206,244,255)(31,187,245,272)(32,204,246,253)(33,185,247,270)(34,202,248,287)(35,183,249,268)(36,200,250,285)(37,81,153,116)(38,98,154,133)(39,79,155,114)(40,96,156,131)(41,77,157,112)(42,94,158,129)(43,75,159,110)(44,92,160,127)(45,73,161,144)(46,90,162,125)(47,107,163,142)(48,88,164,123)(49,105,165,140)(50,86,166,121)(51,103,167,138)(52,84,168,119)(53,101,169,136)(54,82,170,117)(55,99,171,134)(56,80,172,115)(57,97,173,132)(58,78,174,113)(59,95,175,130)(60,76,176,111)(61,93,177,128)(62,74,178,109)(63,91,179,126)(64,108,180,143)(65,89,145,124)(66,106,146,141)(67,87,147,122)(68,104,148,139)(69,85,149,120)(70,102,150,137)(71,83,151,118)(72,100,152,135)>;
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), (1,102,251,137)(2,85,252,120)(3,104,217,139)(4,87,218,122)(5,106,219,141)(6,89,220,124)(7,108,221,143)(8,91,222,126)(9,74,223,109)(10,93,224,128)(11,76,225,111)(12,95,226,130)(13,78,227,113)(14,97,228,132)(15,80,229,115)(16,99,230,134)(17,82,231,117)(18,101,232,136)(19,84,233,119)(20,103,234,138)(21,86,235,121)(22,105,236,140)(23,88,237,123)(24,107,238,142)(25,90,239,125)(26,73,240,144)(27,92,241,127)(28,75,242,110)(29,94,243,129)(30,77,244,112)(31,96,245,131)(32,79,246,114)(33,98,247,133)(34,81,248,116)(35,100,249,135)(36,83,250,118)(37,287,153,202)(38,270,154,185)(39,253,155,204)(40,272,156,187)(41,255,157,206)(42,274,158,189)(43,257,159,208)(44,276,160,191)(45,259,161,210)(46,278,162,193)(47,261,163,212)(48,280,164,195)(49,263,165,214)(50,282,166,197)(51,265,167,216)(52,284,168,199)(53,267,169,182)(54,286,170,201)(55,269,171,184)(56,288,172,203)(57,271,173,186)(58,254,174,205)(59,273,175,188)(60,256,176,207)(61,275,177,190)(62,258,178,209)(63,277,179,192)(64,260,180,211)(65,279,145,194)(66,262,146,213)(67,281,147,196)(68,264,148,215)(69,283,149,198)(70,266,150,181)(71,285,151,200)(72,268,152,183), (1,181,251,266)(2,198,252,283)(3,215,217,264)(4,196,218,281)(5,213,219,262)(6,194,220,279)(7,211,221,260)(8,192,222,277)(9,209,223,258)(10,190,224,275)(11,207,225,256)(12,188,226,273)(13,205,227,254)(14,186,228,271)(15,203,229,288)(16,184,230,269)(17,201,231,286)(18,182,232,267)(19,199,233,284)(20,216,234,265)(21,197,235,282)(22,214,236,263)(23,195,237,280)(24,212,238,261)(25,193,239,278)(26,210,240,259)(27,191,241,276)(28,208,242,257)(29,189,243,274)(30,206,244,255)(31,187,245,272)(32,204,246,253)(33,185,247,270)(34,202,248,287)(35,183,249,268)(36,200,250,285)(37,81,153,116)(38,98,154,133)(39,79,155,114)(40,96,156,131)(41,77,157,112)(42,94,158,129)(43,75,159,110)(44,92,160,127)(45,73,161,144)(46,90,162,125)(47,107,163,142)(48,88,164,123)(49,105,165,140)(50,86,166,121)(51,103,167,138)(52,84,168,119)(53,101,169,136)(54,82,170,117)(55,99,171,134)(56,80,172,115)(57,97,173,132)(58,78,174,113)(59,95,175,130)(60,76,176,111)(61,93,177,128)(62,74,178,109)(63,91,179,126)(64,108,180,143)(65,89,145,124)(66,106,146,141)(67,87,147,122)(68,104,148,139)(69,85,149,120)(70,102,150,137)(71,83,151,118)(72,100,152,135) );
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)], [(1,102,251,137),(2,85,252,120),(3,104,217,139),(4,87,218,122),(5,106,219,141),(6,89,220,124),(7,108,221,143),(8,91,222,126),(9,74,223,109),(10,93,224,128),(11,76,225,111),(12,95,226,130),(13,78,227,113),(14,97,228,132),(15,80,229,115),(16,99,230,134),(17,82,231,117),(18,101,232,136),(19,84,233,119),(20,103,234,138),(21,86,235,121),(22,105,236,140),(23,88,237,123),(24,107,238,142),(25,90,239,125),(26,73,240,144),(27,92,241,127),(28,75,242,110),(29,94,243,129),(30,77,244,112),(31,96,245,131),(32,79,246,114),(33,98,247,133),(34,81,248,116),(35,100,249,135),(36,83,250,118),(37,287,153,202),(38,270,154,185),(39,253,155,204),(40,272,156,187),(41,255,157,206),(42,274,158,189),(43,257,159,208),(44,276,160,191),(45,259,161,210),(46,278,162,193),(47,261,163,212),(48,280,164,195),(49,263,165,214),(50,282,166,197),(51,265,167,216),(52,284,168,199),(53,267,169,182),(54,286,170,201),(55,269,171,184),(56,288,172,203),(57,271,173,186),(58,254,174,205),(59,273,175,188),(60,256,176,207),(61,275,177,190),(62,258,178,209),(63,277,179,192),(64,260,180,211),(65,279,145,194),(66,262,146,213),(67,281,147,196),(68,264,148,215),(69,283,149,198),(70,266,150,181),(71,285,151,200),(72,268,152,183)], [(1,181,251,266),(2,198,252,283),(3,215,217,264),(4,196,218,281),(5,213,219,262),(6,194,220,279),(7,211,221,260),(8,192,222,277),(9,209,223,258),(10,190,224,275),(11,207,225,256),(12,188,226,273),(13,205,227,254),(14,186,228,271),(15,203,229,288),(16,184,230,269),(17,201,231,286),(18,182,232,267),(19,199,233,284),(20,216,234,265),(21,197,235,282),(22,214,236,263),(23,195,237,280),(24,212,238,261),(25,193,239,278),(26,210,240,259),(27,191,241,276),(28,208,242,257),(29,189,243,274),(30,206,244,255),(31,187,245,272),(32,204,246,253),(33,185,247,270),(34,202,248,287),(35,183,249,268),(36,200,250,285),(37,81,153,116),(38,98,154,133),(39,79,155,114),(40,96,156,131),(41,77,157,112),(42,94,158,129),(43,75,159,110),(44,92,160,127),(45,73,161,144),(46,90,162,125),(47,107,163,142),(48,88,164,123),(49,105,165,140),(50,86,166,121),(51,103,167,138),(52,84,168,119),(53,101,169,136),(54,82,170,117),(55,99,171,134),(56,80,172,115),(57,97,173,132),(58,78,174,113),(59,95,175,130),(60,76,176,111),(61,93,177,128),(62,74,178,109),(63,91,179,126),(64,108,180,143),(65,89,145,124),(66,106,146,141),(67,87,147,122),(68,104,148,139),(69,85,149,120),(70,102,150,137),(71,83,151,118),(72,100,152,135)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D6 | D9 | Dic6 | D18 | Dic18 | S3×D4 | S3×Q8 | D4×D9 | Q8×D9 |
| kernel | C36⋊Q8 | C4×Dic9 | Dic9⋊C4 | C4⋊Dic9 | C9×C4⋊C4 | C2×Dic18 | C3×C4⋊C4 | Dic9 | Dic9 | C36 | C2×C12 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C36⋊Q8 ►in GL4(𝔽37) generated by
| 21 | 32 | 0 | 0 |
| 7 | 16 | 0 | 0 |
| 0 | 0 | 11 | 20 |
| 0 | 0 | 17 | 31 |
| 36 | 24 | 0 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 0 | 32 | 10 |
| 0 | 0 | 27 | 5 |
| 16 | 5 | 0 | 0 |
| 30 | 21 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 31 | 6 |
G:=sub<GL(4,GF(37))| [21,7,0,0,32,16,0,0,0,0,11,17,0,0,20,31],[36,3,0,0,24,1,0,0,0,0,32,27,0,0,10,5],[16,30,0,0,5,21,0,0,0,0,31,31,0,0,0,6] >;
C36⋊Q8 in GAP, Magma, Sage, TeX
C_{36}\rtimes Q_8 % in TeX
G:=Group("C36:Q8"); // GroupNames label
G:=SmallGroup(288,98);
// by ID
G=gap.SmallGroup(288,98);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,254,219,58,6725,292,9414]);
// 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=b^-1>;
// generators/relations