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G = C362Q8order 288 = 25·32

1st semidirect product of C36 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C362Q8, C4.4D36, C42Dic18, C36.27D4, C42.4D9, C12.37D12, C12.21Dic6, C91(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.(C122Q8), 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

C1C2×C18 — C362Q8
C1C3C9C18C2×C18C2×Dic9C2×Dic18 — C362Q8
C9C2×C18 — C362Q8
C1C22C42

Generators and relations for C362Q8
 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 [×2], C3, C4 [×6], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C9, Dic3 [×4], C12 [×6], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C18, C18 [×2], Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C4⋊Q8, Dic9 [×4], C36 [×6], C2×C18, C4⋊Dic3 [×4], C4×C12, C2×Dic6 [×2], Dic18 [×4], C2×Dic9 [×4], C2×C36, C2×C36 [×2], C122Q8, C4⋊Dic9 [×4], C4×C36, C2×Dic18 [×2], C362Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], D9, Dic6 [×4], D12 [×2], C22×S3, C4⋊Q8, D18 [×3], C2×Dic6 [×2], C2×D12, Dic18 [×4], D36 [×2], C22×D9, C122Q8, C2×Dic18 [×2], C2×D36, C362Q8

Smallest permutation representation of C362Q8
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 46 214 139)(2 47 215 140)(3 48 216 141)(4 49 181 142)(5 50 182 143)(6 51 183 144)(7 52 184 109)(8 53 185 110)(9 54 186 111)(10 55 187 112)(11 56 188 113)(12 57 189 114)(13 58 190 115)(14 59 191 116)(15 60 192 117)(16 61 193 118)(17 62 194 119)(18 63 195 120)(19 64 196 121)(20 65 197 122)(21 66 198 123)(22 67 199 124)(23 68 200 125)(24 69 201 126)(25 70 202 127)(26 71 203 128)(27 72 204 129)(28 37 205 130)(29 38 206 131)(30 39 207 132)(31 40 208 133)(32 41 209 134)(33 42 210 135)(34 43 211 136)(35 44 212 137)(36 45 213 138)(73 244 167 266)(74 245 168 267)(75 246 169 268)(76 247 170 269)(77 248 171 270)(78 249 172 271)(79 250 173 272)(80 251 174 273)(81 252 175 274)(82 217 176 275)(83 218 177 276)(84 219 178 277)(85 220 179 278)(86 221 180 279)(87 222 145 280)(88 223 146 281)(89 224 147 282)(90 225 148 283)(91 226 149 284)(92 227 150 285)(93 228 151 286)(94 229 152 287)(95 230 153 288)(96 231 154 253)(97 232 155 254)(98 233 156 255)(99 234 157 256)(100 235 158 257)(101 236 159 258)(102 237 160 259)(103 238 161 260)(104 239 162 261)(105 240 163 262)(106 241 164 263)(107 242 165 264)(108 243 166 265)
(1 78 214 172)(2 77 215 171)(3 76 216 170)(4 75 181 169)(5 74 182 168)(6 73 183 167)(7 108 184 166)(8 107 185 165)(9 106 186 164)(10 105 187 163)(11 104 188 162)(12 103 189 161)(13 102 190 160)(14 101 191 159)(15 100 192 158)(16 99 193 157)(17 98 194 156)(18 97 195 155)(19 96 196 154)(20 95 197 153)(21 94 198 152)(22 93 199 151)(23 92 200 150)(24 91 201 149)(25 90 202 148)(26 89 203 147)(27 88 204 146)(28 87 205 145)(29 86 206 180)(30 85 207 179)(31 84 208 178)(32 83 209 177)(33 82 210 176)(34 81 211 175)(35 80 212 174)(36 79 213 173)(37 280 130 222)(38 279 131 221)(39 278 132 220)(40 277 133 219)(41 276 134 218)(42 275 135 217)(43 274 136 252)(44 273 137 251)(45 272 138 250)(46 271 139 249)(47 270 140 248)(48 269 141 247)(49 268 142 246)(50 267 143 245)(51 266 144 244)(52 265 109 243)(53 264 110 242)(54 263 111 241)(55 262 112 240)(56 261 113 239)(57 260 114 238)(58 259 115 237)(59 258 116 236)(60 257 117 235)(61 256 118 234)(62 255 119 233)(63 254 120 232)(64 253 121 231)(65 288 122 230)(66 287 123 229)(67 286 124 228)(68 285 125 227)(69 284 126 226)(70 283 127 225)(71 282 128 224)(72 281 129 223)

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

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

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

78 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J6A6B6C9A9B9C12A···12L18A···18I36A···36AJ
order122234···4444466699912···1218···1836···36
size111122···2363636362222222···22···22···2

78 irreducible representations

dim11112222222222
type++++++-++-++-+
imageC1C2C2C2S3D4Q8D6D9Dic6D12D18Dic18D36
kernelC362Q8C4⋊Dic9C4×C36C2×Dic18C4×C12C36C36C2×C12C42C12C12C2×C4C4C4
# reps1412124338492412

Matrix representation of C362Q8 in GL4(𝔽37) generated by

113100
61700
00429
00812
,
51000
273200
00510
002732
,
283600
8900
003511
00132
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] >;

C362Q8 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

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