direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C36, C42.3C18, C3.(Q8×C12), C4⋊C4.6C18, C4.4(C2×C36), (C4×C36).9C2, C6.19(C6×Q8), C2.2(Q8×C18), (C4×C12).17C6, C36.33(C2×C4), (C2×Q8).7C18, (C6×Q8).27C6, (Q8×C12).2C3, (C3×Q8).8C12, C18.19(C2×Q8), C12.17(C3×Q8), C12.34(C2×C12), C2.5(C22×C36), (Q8×C18).10C2, C18.40(C4○D4), C18.33(C22×C4), C6.33(C22×C12), (C2×C18).74C23, (C2×C36).121C22, C22.7(C22×C18), C2.3(C9×C4○D4), (C9×C4⋊C4).13C2, (C3×C4⋊C4).26C6, C6.40(C3×C4○D4), (C2×C4).19(C2×C18), (C2×C12).139(C2×C6), (C2×C6).79(C22×C6), SmallGroup(288,169)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C9, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C18, C2×C12, C2×C12, C3×Q8, C4×Q8, C36, C36, C2×C18, C4×C12, C3×C4⋊C4, C6×Q8, C2×C36, C2×C36, Q8×C9, Q8×C12, C4×C36, C9×C4⋊C4, Q8×C18, Q8×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, C9, C12, C2×C6, C22×C4, C2×Q8, C4○D4, C18, C2×C12, C3×Q8, C22×C6, C4×Q8, C36, C2×C18, C22×C12, C6×Q8, C3×C4○D4, C2×C36, Q8×C9, C22×C18, Q8×C12, C22×C36, Q8×C18, C9×C4○D4, 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)
(1 121 153 259)(2 122 154 260)(3 123 155 261)(4 124 156 262)(5 125 157 263)(6 126 158 264)(7 127 159 265)(8 128 160 266)(9 129 161 267)(10 130 162 268)(11 131 163 269)(12 132 164 270)(13 133 165 271)(14 134 166 272)(15 135 167 273)(16 136 168 274)(17 137 169 275)(18 138 170 276)(19 139 171 277)(20 140 172 278)(21 141 173 279)(22 142 174 280)(23 143 175 281)(24 144 176 282)(25 109 177 283)(26 110 178 284)(27 111 179 285)(28 112 180 286)(29 113 145 287)(30 114 146 288)(31 115 147 253)(32 116 148 254)(33 117 149 255)(34 118 150 256)(35 119 151 257)(36 120 152 258)(37 200 248 82)(38 201 249 83)(39 202 250 84)(40 203 251 85)(41 204 252 86)(42 205 217 87)(43 206 218 88)(44 207 219 89)(45 208 220 90)(46 209 221 91)(47 210 222 92)(48 211 223 93)(49 212 224 94)(50 213 225 95)(51 214 226 96)(52 215 227 97)(53 216 228 98)(54 181 229 99)(55 182 230 100)(56 183 231 101)(57 184 232 102)(58 185 233 103)(59 186 234 104)(60 187 235 105)(61 188 236 106)(62 189 237 107)(63 190 238 108)(64 191 239 73)(65 192 240 74)(66 193 241 75)(67 194 242 76)(68 195 243 77)(69 196 244 78)(70 197 245 79)(71 198 246 80)(72 199 247 81)
(1 252 153 41)(2 217 154 42)(3 218 155 43)(4 219 156 44)(5 220 157 45)(6 221 158 46)(7 222 159 47)(8 223 160 48)(9 224 161 49)(10 225 162 50)(11 226 163 51)(12 227 164 52)(13 228 165 53)(14 229 166 54)(15 230 167 55)(16 231 168 56)(17 232 169 57)(18 233 170 58)(19 234 171 59)(20 235 172 60)(21 236 173 61)(22 237 174 62)(23 238 175 63)(24 239 176 64)(25 240 177 65)(26 241 178 66)(27 242 179 67)(28 243 180 68)(29 244 145 69)(30 245 146 70)(31 246 147 71)(32 247 148 72)(33 248 149 37)(34 249 150 38)(35 250 151 39)(36 251 152 40)(73 144 191 282)(74 109 192 283)(75 110 193 284)(76 111 194 285)(77 112 195 286)(78 113 196 287)(79 114 197 288)(80 115 198 253)(81 116 199 254)(82 117 200 255)(83 118 201 256)(84 119 202 257)(85 120 203 258)(86 121 204 259)(87 122 205 260)(88 123 206 261)(89 124 207 262)(90 125 208 263)(91 126 209 264)(92 127 210 265)(93 128 211 266)(94 129 212 267)(95 130 213 268)(96 131 214 269)(97 132 215 270)(98 133 216 271)(99 134 181 272)(100 135 182 273)(101 136 183 274)(102 137 184 275)(103 138 185 276)(104 139 186 277)(105 140 187 278)(106 141 188 279)(107 142 189 280)(108 143 190 281)
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), (1,121,153,259)(2,122,154,260)(3,123,155,261)(4,124,156,262)(5,125,157,263)(6,126,158,264)(7,127,159,265)(8,128,160,266)(9,129,161,267)(10,130,162,268)(11,131,163,269)(12,132,164,270)(13,133,165,271)(14,134,166,272)(15,135,167,273)(16,136,168,274)(17,137,169,275)(18,138,170,276)(19,139,171,277)(20,140,172,278)(21,141,173,279)(22,142,174,280)(23,143,175,281)(24,144,176,282)(25,109,177,283)(26,110,178,284)(27,111,179,285)(28,112,180,286)(29,113,145,287)(30,114,146,288)(31,115,147,253)(32,116,148,254)(33,117,149,255)(34,118,150,256)(35,119,151,257)(36,120,152,258)(37,200,248,82)(38,201,249,83)(39,202,250,84)(40,203,251,85)(41,204,252,86)(42,205,217,87)(43,206,218,88)(44,207,219,89)(45,208,220,90)(46,209,221,91)(47,210,222,92)(48,211,223,93)(49,212,224,94)(50,213,225,95)(51,214,226,96)(52,215,227,97)(53,216,228,98)(54,181,229,99)(55,182,230,100)(56,183,231,101)(57,184,232,102)(58,185,233,103)(59,186,234,104)(60,187,235,105)(61,188,236,106)(62,189,237,107)(63,190,238,108)(64,191,239,73)(65,192,240,74)(66,193,241,75)(67,194,242,76)(68,195,243,77)(69,196,244,78)(70,197,245,79)(71,198,246,80)(72,199,247,81), (1,252,153,41)(2,217,154,42)(3,218,155,43)(4,219,156,44)(5,220,157,45)(6,221,158,46)(7,222,159,47)(8,223,160,48)(9,224,161,49)(10,225,162,50)(11,226,163,51)(12,227,164,52)(13,228,165,53)(14,229,166,54)(15,230,167,55)(16,231,168,56)(17,232,169,57)(18,233,170,58)(19,234,171,59)(20,235,172,60)(21,236,173,61)(22,237,174,62)(23,238,175,63)(24,239,176,64)(25,240,177,65)(26,241,178,66)(27,242,179,67)(28,243,180,68)(29,244,145,69)(30,245,146,70)(31,246,147,71)(32,247,148,72)(33,248,149,37)(34,249,150,38)(35,250,151,39)(36,251,152,40)(73,144,191,282)(74,109,192,283)(75,110,193,284)(76,111,194,285)(77,112,195,286)(78,113,196,287)(79,114,197,288)(80,115,198,253)(81,116,199,254)(82,117,200,255)(83,118,201,256)(84,119,202,257)(85,120,203,258)(86,121,204,259)(87,122,205,260)(88,123,206,261)(89,124,207,262)(90,125,208,263)(91,126,209,264)(92,127,210,265)(93,128,211,266)(94,129,212,267)(95,130,213,268)(96,131,214,269)(97,132,215,270)(98,133,216,271)(99,134,181,272)(100,135,182,273)(101,136,183,274)(102,137,184,275)(103,138,185,276)(104,139,186,277)(105,140,187,278)(106,141,188,279)(107,142,189,280)(108,143,190,281)>;
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), (1,121,153,259)(2,122,154,260)(3,123,155,261)(4,124,156,262)(5,125,157,263)(6,126,158,264)(7,127,159,265)(8,128,160,266)(9,129,161,267)(10,130,162,268)(11,131,163,269)(12,132,164,270)(13,133,165,271)(14,134,166,272)(15,135,167,273)(16,136,168,274)(17,137,169,275)(18,138,170,276)(19,139,171,277)(20,140,172,278)(21,141,173,279)(22,142,174,280)(23,143,175,281)(24,144,176,282)(25,109,177,283)(26,110,178,284)(27,111,179,285)(28,112,180,286)(29,113,145,287)(30,114,146,288)(31,115,147,253)(32,116,148,254)(33,117,149,255)(34,118,150,256)(35,119,151,257)(36,120,152,258)(37,200,248,82)(38,201,249,83)(39,202,250,84)(40,203,251,85)(41,204,252,86)(42,205,217,87)(43,206,218,88)(44,207,219,89)(45,208,220,90)(46,209,221,91)(47,210,222,92)(48,211,223,93)(49,212,224,94)(50,213,225,95)(51,214,226,96)(52,215,227,97)(53,216,228,98)(54,181,229,99)(55,182,230,100)(56,183,231,101)(57,184,232,102)(58,185,233,103)(59,186,234,104)(60,187,235,105)(61,188,236,106)(62,189,237,107)(63,190,238,108)(64,191,239,73)(65,192,240,74)(66,193,241,75)(67,194,242,76)(68,195,243,77)(69,196,244,78)(70,197,245,79)(71,198,246,80)(72,199,247,81), (1,252,153,41)(2,217,154,42)(3,218,155,43)(4,219,156,44)(5,220,157,45)(6,221,158,46)(7,222,159,47)(8,223,160,48)(9,224,161,49)(10,225,162,50)(11,226,163,51)(12,227,164,52)(13,228,165,53)(14,229,166,54)(15,230,167,55)(16,231,168,56)(17,232,169,57)(18,233,170,58)(19,234,171,59)(20,235,172,60)(21,236,173,61)(22,237,174,62)(23,238,175,63)(24,239,176,64)(25,240,177,65)(26,241,178,66)(27,242,179,67)(28,243,180,68)(29,244,145,69)(30,245,146,70)(31,246,147,71)(32,247,148,72)(33,248,149,37)(34,249,150,38)(35,250,151,39)(36,251,152,40)(73,144,191,282)(74,109,192,283)(75,110,193,284)(76,111,194,285)(77,112,195,286)(78,113,196,287)(79,114,197,288)(80,115,198,253)(81,116,199,254)(82,117,200,255)(83,118,201,256)(84,119,202,257)(85,120,203,258)(86,121,204,259)(87,122,205,260)(88,123,206,261)(89,124,207,262)(90,125,208,263)(91,126,209,264)(92,127,210,265)(93,128,211,266)(94,129,212,267)(95,130,213,268)(96,131,214,269)(97,132,215,270)(98,133,216,271)(99,134,181,272)(100,135,182,273)(101,136,183,274)(102,137,184,275)(103,138,185,276)(104,139,186,277)(105,140,187,278)(106,141,188,279)(107,142,189,280)(108,143,190,281) );
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)], [(1,121,153,259),(2,122,154,260),(3,123,155,261),(4,124,156,262),(5,125,157,263),(6,126,158,264),(7,127,159,265),(8,128,160,266),(9,129,161,267),(10,130,162,268),(11,131,163,269),(12,132,164,270),(13,133,165,271),(14,134,166,272),(15,135,167,273),(16,136,168,274),(17,137,169,275),(18,138,170,276),(19,139,171,277),(20,140,172,278),(21,141,173,279),(22,142,174,280),(23,143,175,281),(24,144,176,282),(25,109,177,283),(26,110,178,284),(27,111,179,285),(28,112,180,286),(29,113,145,287),(30,114,146,288),(31,115,147,253),(32,116,148,254),(33,117,149,255),(34,118,150,256),(35,119,151,257),(36,120,152,258),(37,200,248,82),(38,201,249,83),(39,202,250,84),(40,203,251,85),(41,204,252,86),(42,205,217,87),(43,206,218,88),(44,207,219,89),(45,208,220,90),(46,209,221,91),(47,210,222,92),(48,211,223,93),(49,212,224,94),(50,213,225,95),(51,214,226,96),(52,215,227,97),(53,216,228,98),(54,181,229,99),(55,182,230,100),(56,183,231,101),(57,184,232,102),(58,185,233,103),(59,186,234,104),(60,187,235,105),(61,188,236,106),(62,189,237,107),(63,190,238,108),(64,191,239,73),(65,192,240,74),(66,193,241,75),(67,194,242,76),(68,195,243,77),(69,196,244,78),(70,197,245,79),(71,198,246,80),(72,199,247,81)], [(1,252,153,41),(2,217,154,42),(3,218,155,43),(4,219,156,44),(5,220,157,45),(6,221,158,46),(7,222,159,47),(8,223,160,48),(9,224,161,49),(10,225,162,50),(11,226,163,51),(12,227,164,52),(13,228,165,53),(14,229,166,54),(15,230,167,55),(16,231,168,56),(17,232,169,57),(18,233,170,58),(19,234,171,59),(20,235,172,60),(21,236,173,61),(22,237,174,62),(23,238,175,63),(24,239,176,64),(25,240,177,65),(26,241,178,66),(27,242,179,67),(28,243,180,68),(29,244,145,69),(30,245,146,70),(31,246,147,71),(32,247,148,72),(33,248,149,37),(34,249,150,38),(35,250,151,39),(36,251,152,40),(73,144,191,282),(74,109,192,283),(75,110,193,284),(76,111,194,285),(77,112,195,286),(78,113,196,287),(79,114,197,288),(80,115,198,253),(81,116,199,254),(82,117,200,255),(83,118,201,256),(84,119,202,257),(85,120,203,258),(86,121,204,259),(87,122,205,260),(88,123,206,261),(89,124,207,262),(90,125,208,263),(91,126,209,264),(92,127,210,265),(93,128,211,266),(94,129,212,267),(95,130,213,268),(96,131,214,269),(97,132,215,270),(98,133,216,271),(99,134,181,272),(100,135,182,273),(101,136,183,274),(102,137,184,275),(103,138,185,276),(104,139,186,277),(105,140,187,278),(106,141,188,279),(107,142,189,280),(108,143,190,281)]])
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