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## G = Q8×C36order 288 = 25·32

### Direct product of C36 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C36
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C9×C4⋊C4 — Q8×C36
 Lower central C1 — C2 — Q8×C36
 Upper central C1 — C2×C36 — Q8×C36

Generators and relations for Q8×C36
G = < a,b,c | a36=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 114 in 105 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C3, C4 [×8], C4 [×3], C22, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C9, C12 [×8], C12 [×3], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C18 [×3], C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C36 [×8], C36 [×3], C2×C18, C4×C12 [×3], C3×C4⋊C4 [×3], C6×Q8, C2×C36, C2×C36 [×6], Q8×C9 [×4], Q8×C12, C4×C36 [×3], C9×C4⋊C4 [×3], Q8×C18, Q8×C36
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], Q8 [×2], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, C18 [×7], C2×C12 [×6], C3×Q8 [×2], C22×C6, C4×Q8, C36 [×4], C2×C18 [×7], C22×C12, C6×Q8, C3×C4○D4, C2×C36 [×6], Q8×C9 [×2], C22×C18, Q8×C12, C22×C36, Q8×C18, C9×C4○D4, Q8×C36

Smallest permutation representation of Q8×C36
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)
(1 203 267 117)(2 204 268 118)(3 205 269 119)(4 206 270 120)(5 207 271 121)(6 208 272 122)(7 209 273 123)(8 210 274 124)(9 211 275 125)(10 212 276 126)(11 213 277 127)(12 214 278 128)(13 215 279 129)(14 216 280 130)(15 181 281 131)(16 182 282 132)(17 183 283 133)(18 184 284 134)(19 185 285 135)(20 186 286 136)(21 187 287 137)(22 188 288 138)(23 189 253 139)(24 190 254 140)(25 191 255 141)(26 192 256 142)(27 193 257 143)(28 194 258 144)(29 195 259 109)(30 196 260 110)(31 197 261 111)(32 198 262 112)(33 199 263 113)(34 200 264 114)(35 201 265 115)(36 202 266 116)(37 73 239 171)(38 74 240 172)(39 75 241 173)(40 76 242 174)(41 77 243 175)(42 78 244 176)(43 79 245 177)(44 80 246 178)(45 81 247 179)(46 82 248 180)(47 83 249 145)(48 84 250 146)(49 85 251 147)(50 86 252 148)(51 87 217 149)(52 88 218 150)(53 89 219 151)(54 90 220 152)(55 91 221 153)(56 92 222 154)(57 93 223 155)(58 94 224 156)(59 95 225 157)(60 96 226 158)(61 97 227 159)(62 98 228 160)(63 99 229 161)(64 100 230 162)(65 101 231 163)(66 102 232 164)(67 103 233 165)(68 104 234 166)(69 105 235 167)(70 106 236 168)(71 107 237 169)(72 108 238 170)
(1 168 267 106)(2 169 268 107)(3 170 269 108)(4 171 270 73)(5 172 271 74)(6 173 272 75)(7 174 273 76)(8 175 274 77)(9 176 275 78)(10 177 276 79)(11 178 277 80)(12 179 278 81)(13 180 279 82)(14 145 280 83)(15 146 281 84)(16 147 282 85)(17 148 283 86)(18 149 284 87)(19 150 285 88)(20 151 286 89)(21 152 287 90)(22 153 288 91)(23 154 253 92)(24 155 254 93)(25 156 255 94)(26 157 256 95)(27 158 257 96)(28 159 258 97)(29 160 259 98)(30 161 260 99)(31 162 261 100)(32 163 262 101)(33 164 263 102)(34 165 264 103)(35 166 265 104)(36 167 266 105)(37 206 239 120)(38 207 240 121)(39 208 241 122)(40 209 242 123)(41 210 243 124)(42 211 244 125)(43 212 245 126)(44 213 246 127)(45 214 247 128)(46 215 248 129)(47 216 249 130)(48 181 250 131)(49 182 251 132)(50 183 252 133)(51 184 217 134)(52 185 218 135)(53 186 219 136)(54 187 220 137)(55 188 221 138)(56 189 222 139)(57 190 223 140)(58 191 224 141)(59 192 225 142)(60 193 226 143)(61 194 227 144)(62 195 228 109)(63 196 229 110)(64 197 230 111)(65 198 231 112)(66 199 232 113)(67 200 233 114)(68 201 234 115)(69 202 235 116)(70 203 236 117)(71 204 237 118)(72 205 238 119)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4P 6A ··· 6F 9A ··· 9F 12A ··· 12H 12I ··· 12AF 18A ··· 18R 36A ··· 36X 36Y ··· 36CR order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C9 C12 C18 C18 C18 C36 Q8 C4○D4 C3×Q8 C3×C4○D4 Q8×C9 C9×C4○D4 kernel Q8×C36 C4×C36 C9×C4⋊C4 Q8×C18 Q8×C12 Q8×C9 C4×C12 C3×C4⋊C4 C6×Q8 C4×Q8 C3×Q8 C42 C4⋊C4 C2×Q8 Q8 C36 C18 C12 C6 C4 C2 # reps 1 3 3 1 2 8 6 6 2 6 16 18 18 6 48 2 2 4 4 12 12

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

 28 0 0 0 0 31 0 0 0 0 8 0 0 0 0 8
,
 36 0 0 0 0 1 0 0 0 0 1 2 0 0 36 36
,
 36 0 0 0 0 36 0 0 0 0 18 27 0 0 14 19
G:=sub<GL(4,GF(37))| [28,0,0,0,0,31,0,0,0,0,8,0,0,0,0,8],[36,0,0,0,0,1,0,0,0,0,1,36,0,0,2,36],[36,0,0,0,0,36,0,0,0,0,18,14,0,0,27,19] >;

Q8×C36 in GAP, Magma, Sage, TeX

Q_8\times C_{36}
% in TeX

G:=Group("Q8xC36");
// GroupNames label

G:=SmallGroup(288,169);
// by ID

G=gap.SmallGroup(288,169);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,176,394,360]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽