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G = C36⋊Q8order 288 = 25·32

The semidirect product of C36 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36⋊Q8, Dic91Q8, C41Dic18, Dic9.2D4, C12.3Dic6, C92(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

C1C2×C18 — C36⋊Q8
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — C36⋊Q8
C9C2×C18 — C36⋊Q8
C1C22C4⋊C4

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 [×3], C3, C4 [×2], C4 [×8], C22, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C9, Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], C18 [×3], Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C4⋊Q8, Dic9 [×4], Dic9 [×2], C36 [×2], C36 [×2], C2×C18, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], Dic18 [×4], C2×Dic9 [×2], C2×Dic9 [×2], C2×C36, C2×C36 [×2], C12⋊Q8, C4×Dic9, Dic9⋊C4 [×2], C4⋊Dic9, C9×C4⋊C4, C2×Dic18 [×2], C36⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], D9, Dic6 [×2], C22×S3, C4⋊Q8, D18 [×3], C2×Dic6, S3×D4, S3×Q8, Dic18 [×2], C22×D9, C12⋊Q8, C2×Dic18, D4×D9, Q8×D9, C36⋊Q8

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)
(1 283 97 120)(2 266 98 139)(3 285 99 122)(4 268 100 141)(5 287 101 124)(6 270 102 143)(7 253 103 126)(8 272 104 109)(9 255 105 128)(10 274 106 111)(11 257 107 130)(12 276 108 113)(13 259 73 132)(14 278 74 115)(15 261 75 134)(16 280 76 117)(17 263 77 136)(18 282 78 119)(19 265 79 138)(20 284 80 121)(21 267 81 140)(22 286 82 123)(23 269 83 142)(24 288 84 125)(25 271 85 144)(26 254 86 127)(27 273 87 110)(28 256 88 129)(29 275 89 112)(30 258 90 131)(31 277 91 114)(32 260 92 133)(33 279 93 116)(34 262 94 135)(35 281 95 118)(36 264 96 137)(37 167 238 198)(38 150 239 181)(39 169 240 200)(40 152 241 183)(41 171 242 202)(42 154 243 185)(43 173 244 204)(44 156 245 187)(45 175 246 206)(46 158 247 189)(47 177 248 208)(48 160 249 191)(49 179 250 210)(50 162 251 193)(51 145 252 212)(52 164 217 195)(53 147 218 214)(54 166 219 197)(55 149 220 216)(56 168 221 199)(57 151 222 182)(58 170 223 201)(59 153 224 184)(60 172 225 203)(61 155 226 186)(62 174 227 205)(63 157 228 188)(64 176 229 207)(65 159 230 190)(66 178 231 209)(67 161 232 192)(68 180 233 211)(69 163 234 194)(70 146 235 213)(71 165 236 196)(72 148 237 215)
(1 221 97 56)(2 238 98 37)(3 219 99 54)(4 236 100 71)(5 217 101 52)(6 234 102 69)(7 251 103 50)(8 232 104 67)(9 249 105 48)(10 230 106 65)(11 247 107 46)(12 228 108 63)(13 245 73 44)(14 226 74 61)(15 243 75 42)(16 224 76 59)(17 241 77 40)(18 222 78 57)(19 239 79 38)(20 220 80 55)(21 237 81 72)(22 218 82 53)(23 235 83 70)(24 252 84 51)(25 233 85 68)(26 250 86 49)(27 231 87 66)(28 248 88 47)(29 229 89 64)(30 246 90 45)(31 227 91 62)(32 244 92 43)(33 225 93 60)(34 242 94 41)(35 223 95 58)(36 240 96 39)(109 192 272 161)(110 209 273 178)(111 190 274 159)(112 207 275 176)(113 188 276 157)(114 205 277 174)(115 186 278 155)(116 203 279 172)(117 184 280 153)(118 201 281 170)(119 182 282 151)(120 199 283 168)(121 216 284 149)(122 197 285 166)(123 214 286 147)(124 195 287 164)(125 212 288 145)(126 193 253 162)(127 210 254 179)(128 191 255 160)(129 208 256 177)(130 189 257 158)(131 206 258 175)(132 187 259 156)(133 204 260 173)(134 185 261 154)(135 202 262 171)(136 183 263 152)(137 200 264 169)(138 181 265 150)(139 198 266 167)(140 215 267 148)(141 196 268 165)(142 213 269 146)(143 194 270 163)(144 211 271 180)

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), (1,283,97,120)(2,266,98,139)(3,285,99,122)(4,268,100,141)(5,287,101,124)(6,270,102,143)(7,253,103,126)(8,272,104,109)(9,255,105,128)(10,274,106,111)(11,257,107,130)(12,276,108,113)(13,259,73,132)(14,278,74,115)(15,261,75,134)(16,280,76,117)(17,263,77,136)(18,282,78,119)(19,265,79,138)(20,284,80,121)(21,267,81,140)(22,286,82,123)(23,269,83,142)(24,288,84,125)(25,271,85,144)(26,254,86,127)(27,273,87,110)(28,256,88,129)(29,275,89,112)(30,258,90,131)(31,277,91,114)(32,260,92,133)(33,279,93,116)(34,262,94,135)(35,281,95,118)(36,264,96,137)(37,167,238,198)(38,150,239,181)(39,169,240,200)(40,152,241,183)(41,171,242,202)(42,154,243,185)(43,173,244,204)(44,156,245,187)(45,175,246,206)(46,158,247,189)(47,177,248,208)(48,160,249,191)(49,179,250,210)(50,162,251,193)(51,145,252,212)(52,164,217,195)(53,147,218,214)(54,166,219,197)(55,149,220,216)(56,168,221,199)(57,151,222,182)(58,170,223,201)(59,153,224,184)(60,172,225,203)(61,155,226,186)(62,174,227,205)(63,157,228,188)(64,176,229,207)(65,159,230,190)(66,178,231,209)(67,161,232,192)(68,180,233,211)(69,163,234,194)(70,146,235,213)(71,165,236,196)(72,148,237,215), (1,221,97,56)(2,238,98,37)(3,219,99,54)(4,236,100,71)(5,217,101,52)(6,234,102,69)(7,251,103,50)(8,232,104,67)(9,249,105,48)(10,230,106,65)(11,247,107,46)(12,228,108,63)(13,245,73,44)(14,226,74,61)(15,243,75,42)(16,224,76,59)(17,241,77,40)(18,222,78,57)(19,239,79,38)(20,220,80,55)(21,237,81,72)(22,218,82,53)(23,235,83,70)(24,252,84,51)(25,233,85,68)(26,250,86,49)(27,231,87,66)(28,248,88,47)(29,229,89,64)(30,246,90,45)(31,227,91,62)(32,244,92,43)(33,225,93,60)(34,242,94,41)(35,223,95,58)(36,240,96,39)(109,192,272,161)(110,209,273,178)(111,190,274,159)(112,207,275,176)(113,188,276,157)(114,205,277,174)(115,186,278,155)(116,203,279,172)(117,184,280,153)(118,201,281,170)(119,182,282,151)(120,199,283,168)(121,216,284,149)(122,197,285,166)(123,214,286,147)(124,195,287,164)(125,212,288,145)(126,193,253,162)(127,210,254,179)(128,191,255,160)(129,208,256,177)(130,189,257,158)(131,206,258,175)(132,187,259,156)(133,204,260,173)(134,185,261,154)(135,202,262,171)(136,183,263,152)(137,200,264,169)(138,181,265,150)(139,198,266,167)(140,215,267,148)(141,196,268,165)(142,213,269,146)(143,194,270,163)(144,211,271,180)>;

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), (1,283,97,120)(2,266,98,139)(3,285,99,122)(4,268,100,141)(5,287,101,124)(6,270,102,143)(7,253,103,126)(8,272,104,109)(9,255,105,128)(10,274,106,111)(11,257,107,130)(12,276,108,113)(13,259,73,132)(14,278,74,115)(15,261,75,134)(16,280,76,117)(17,263,77,136)(18,282,78,119)(19,265,79,138)(20,284,80,121)(21,267,81,140)(22,286,82,123)(23,269,83,142)(24,288,84,125)(25,271,85,144)(26,254,86,127)(27,273,87,110)(28,256,88,129)(29,275,89,112)(30,258,90,131)(31,277,91,114)(32,260,92,133)(33,279,93,116)(34,262,94,135)(35,281,95,118)(36,264,96,137)(37,167,238,198)(38,150,239,181)(39,169,240,200)(40,152,241,183)(41,171,242,202)(42,154,243,185)(43,173,244,204)(44,156,245,187)(45,175,246,206)(46,158,247,189)(47,177,248,208)(48,160,249,191)(49,179,250,210)(50,162,251,193)(51,145,252,212)(52,164,217,195)(53,147,218,214)(54,166,219,197)(55,149,220,216)(56,168,221,199)(57,151,222,182)(58,170,223,201)(59,153,224,184)(60,172,225,203)(61,155,226,186)(62,174,227,205)(63,157,228,188)(64,176,229,207)(65,159,230,190)(66,178,231,209)(67,161,232,192)(68,180,233,211)(69,163,234,194)(70,146,235,213)(71,165,236,196)(72,148,237,215), (1,221,97,56)(2,238,98,37)(3,219,99,54)(4,236,100,71)(5,217,101,52)(6,234,102,69)(7,251,103,50)(8,232,104,67)(9,249,105,48)(10,230,106,65)(11,247,107,46)(12,228,108,63)(13,245,73,44)(14,226,74,61)(15,243,75,42)(16,224,76,59)(17,241,77,40)(18,222,78,57)(19,239,79,38)(20,220,80,55)(21,237,81,72)(22,218,82,53)(23,235,83,70)(24,252,84,51)(25,233,85,68)(26,250,86,49)(27,231,87,66)(28,248,88,47)(29,229,89,64)(30,246,90,45)(31,227,91,62)(32,244,92,43)(33,225,93,60)(34,242,94,41)(35,223,95,58)(36,240,96,39)(109,192,272,161)(110,209,273,178)(111,190,274,159)(112,207,275,176)(113,188,276,157)(114,205,277,174)(115,186,278,155)(116,203,279,172)(117,184,280,153)(118,201,281,170)(119,182,282,151)(120,199,283,168)(121,216,284,149)(122,197,285,166)(123,214,286,147)(124,195,287,164)(125,212,288,145)(126,193,253,162)(127,210,254,179)(128,191,255,160)(129,208,256,177)(130,189,257,158)(131,206,258,175)(132,187,259,156)(133,204,260,173)(134,185,261,154)(135,202,262,171)(136,183,263,152)(137,200,264,169)(138,181,265,150)(139,198,266,167)(140,215,267,148)(141,196,268,165)(142,213,269,146)(143,194,270,163)(144,211,271,180) );

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)], [(1,283,97,120),(2,266,98,139),(3,285,99,122),(4,268,100,141),(5,287,101,124),(6,270,102,143),(7,253,103,126),(8,272,104,109),(9,255,105,128),(10,274,106,111),(11,257,107,130),(12,276,108,113),(13,259,73,132),(14,278,74,115),(15,261,75,134),(16,280,76,117),(17,263,77,136),(18,282,78,119),(19,265,79,138),(20,284,80,121),(21,267,81,140),(22,286,82,123),(23,269,83,142),(24,288,84,125),(25,271,85,144),(26,254,86,127),(27,273,87,110),(28,256,88,129),(29,275,89,112),(30,258,90,131),(31,277,91,114),(32,260,92,133),(33,279,93,116),(34,262,94,135),(35,281,95,118),(36,264,96,137),(37,167,238,198),(38,150,239,181),(39,169,240,200),(40,152,241,183),(41,171,242,202),(42,154,243,185),(43,173,244,204),(44,156,245,187),(45,175,246,206),(46,158,247,189),(47,177,248,208),(48,160,249,191),(49,179,250,210),(50,162,251,193),(51,145,252,212),(52,164,217,195),(53,147,218,214),(54,166,219,197),(55,149,220,216),(56,168,221,199),(57,151,222,182),(58,170,223,201),(59,153,224,184),(60,172,225,203),(61,155,226,186),(62,174,227,205),(63,157,228,188),(64,176,229,207),(65,159,230,190),(66,178,231,209),(67,161,232,192),(68,180,233,211),(69,163,234,194),(70,146,235,213),(71,165,236,196),(72,148,237,215)], [(1,221,97,56),(2,238,98,37),(3,219,99,54),(4,236,100,71),(5,217,101,52),(6,234,102,69),(7,251,103,50),(8,232,104,67),(9,249,105,48),(10,230,106,65),(11,247,107,46),(12,228,108,63),(13,245,73,44),(14,226,74,61),(15,243,75,42),(16,224,76,59),(17,241,77,40),(18,222,78,57),(19,239,79,38),(20,220,80,55),(21,237,81,72),(22,218,82,53),(23,235,83,70),(24,252,84,51),(25,233,85,68),(26,250,86,49),(27,231,87,66),(28,248,88,47),(29,229,89,64),(30,246,90,45),(31,227,91,62),(32,244,92,43),(33,225,93,60),(34,242,94,41),(35,223,95,58),(36,240,96,39),(109,192,272,161),(110,209,273,178),(111,190,274,159),(112,207,275,176),(113,188,276,157),(114,205,277,174),(115,186,278,155),(116,203,279,172),(117,184,280,153),(118,201,281,170),(119,182,282,151),(120,199,283,168),(121,216,284,149),(122,197,285,166),(123,214,286,147),(124,195,287,164),(125,212,288,145),(126,193,253,162),(127,210,254,179),(128,191,255,160),(129,208,256,177),(130,189,257,158),(131,206,258,175),(132,187,259,156),(133,204,260,173),(134,185,261,154),(135,202,262,171),(136,183,263,152),(137,200,264,169),(138,181,265,150),(139,198,266,167),(140,215,267,148),(141,196,268,165),(142,213,269,146),(143,194,270,163),(144,211,271,180)])

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12223444444444466699912···1218···1836···36
size1111222441818181836362222224···42···24···4

54 irreducible representations

dim1111112222222224444
type++++++++--++-+-+-+-
imageC1C2C2C2C2C2S3D4Q8Q8D6D9Dic6D18Dic18S3×D4S3×Q8D4×D9Q8×D9
kernelC36⋊Q8C4×Dic9Dic9⋊C4C4⋊Dic9C9×C4⋊C4C2×Dic18C3×C4⋊C4Dic9Dic9C36C2×C12C4⋊C4C12C2×C4C4C6C6C2C2
# reps11211212223349121133

Matrix representation of C36⋊Q8 in GL4(𝔽37) generated by

213200
71600
001120
001731
,
362400
3100
003210
00275
,
16500
302100
00310
00316
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

׿
×
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