direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C40, C4.4(C2×C40), (C4×C8).3C10, (C4×C40).6C2, C4⋊C4.12C20, C4⋊C8.11C10, C2.2(Q8×C20), C20.67(C2×C8), (C2×Q8).9C20, C4.24(Q8×C10), C10.42(C4×Q8), C2.5(C22×C40), (Q8×C10).29C4, (C4×Q8).11C10, (Q8×C20).24C2, C20.130(C2×Q8), C10.75(C8○D4), C10.58(C22×C8), C42.74(C2×C10), C20.356(C4○D4), (C4×C20).359C22, (C2×C20).993C23, (C2×C40).362C22, C22.23(C22×C20), C2.3(C5×C8○D4), (C5×C4⋊C8).24C2, (C5×C4⋊C4).37C4, C4.54(C5×C4○D4), (C2×C4).37(C2×C20), (C2×C20).387(C2×C4), (C2×C8).108(C2×C10), (C2×C10).344(C22×C4), (C2×C4).161(C22×C10), SmallGroup(320,946)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C40
G = < a,b,c | a40=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 114 in 102 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C20, C20, C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C40, C40, C2×C20, C2×C20, C5×Q8, C8×Q8, C4×C20, C5×C4⋊C4, C2×C40, C2×C40, Q8×C10, C4×C40, C5×C4⋊C8, Q8×C20, Q8×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, Q8, C23, C10, C2×C8, C22×C4, C2×Q8, C4○D4, C20, C2×C10, C4×Q8, C22×C8, C8○D4, C40, C2×C20, C5×Q8, C22×C10, C8×Q8, C2×C40, C22×C20, Q8×C10, C5×C4○D4, Q8×C20, C22×C40, C5×C8○D4, Q8×C40
►Regular action on 320 points
Generators in S320
(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)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 83 254 229)(2 84 255 230)(3 85 256 231)(4 86 257 232)(5 87 258 233)(6 88 259 234)(7 89 260 235)(8 90 261 236)(9 91 262 237)(10 92 263 238)(11 93 264 239)(12 94 265 240)(13 95 266 201)(14 96 267 202)(15 97 268 203)(16 98 269 204)(17 99 270 205)(18 100 271 206)(19 101 272 207)(20 102 273 208)(21 103 274 209)(22 104 275 210)(23 105 276 211)(24 106 277 212)(25 107 278 213)(26 108 279 214)(27 109 280 215)(28 110 241 216)(29 111 242 217)(30 112 243 218)(31 113 244 219)(32 114 245 220)(33 115 246 221)(34 116 247 222)(35 117 248 223)(36 118 249 224)(37 119 250 225)(38 120 251 226)(39 81 252 227)(40 82 253 228)(41 145 196 313)(42 146 197 314)(43 147 198 315)(44 148 199 316)(45 149 200 317)(46 150 161 318)(47 151 162 319)(48 152 163 320)(49 153 164 281)(50 154 165 282)(51 155 166 283)(52 156 167 284)(53 157 168 285)(54 158 169 286)(55 159 170 287)(56 160 171 288)(57 121 172 289)(58 122 173 290)(59 123 174 291)(60 124 175 292)(61 125 176 293)(62 126 177 294)(63 127 178 295)(64 128 179 296)(65 129 180 297)(66 130 181 298)(67 131 182 299)(68 132 183 300)(69 133 184 301)(70 134 185 302)(71 135 186 303)(72 136 187 304)(73 137 188 305)(74 138 189 306)(75 139 190 307)(76 140 191 308)(77 141 192 309)(78 142 193 310)(79 143 194 311)(80 144 195 312)
(1 179 254 64)(2 180 255 65)(3 181 256 66)(4 182 257 67)(5 183 258 68)(6 184 259 69)(7 185 260 70)(8 186 261 71)(9 187 262 72)(10 188 263 73)(11 189 264 74)(12 190 265 75)(13 191 266 76)(14 192 267 77)(15 193 268 78)(16 194 269 79)(17 195 270 80)(18 196 271 41)(19 197 272 42)(20 198 273 43)(21 199 274 44)(22 200 275 45)(23 161 276 46)(24 162 277 47)(25 163 278 48)(26 164 279 49)(27 165 280 50)(28 166 241 51)(29 167 242 52)(30 168 243 53)(31 169 244 54)(32 170 245 55)(33 171 246 56)(34 172 247 57)(35 173 248 58)(36 174 249 59)(37 175 250 60)(38 176 251 61)(39 177 252 62)(40 178 253 63)(81 126 227 294)(82 127 228 295)(83 128 229 296)(84 129 230 297)(85 130 231 298)(86 131 232 299)(87 132 233 300)(88 133 234 301)(89 134 235 302)(90 135 236 303)(91 136 237 304)(92 137 238 305)(93 138 239 306)(94 139 240 307)(95 140 201 308)(96 141 202 309)(97 142 203 310)(98 143 204 311)(99 144 205 312)(100 145 206 313)(101 146 207 314)(102 147 208 315)(103 148 209 316)(104 149 210 317)(105 150 211 318)(106 151 212 319)(107 152 213 320)(108 153 214 281)(109 154 215 282)(110 155 216 283)(111 156 217 284)(112 157 218 285)(113 158 219 286)(114 159 220 287)(115 160 221 288)(116 121 222 289)(117 122 223 290)(118 123 224 291)(119 124 225 292)(120 125 226 293)
G:=sub<Sym(320)| (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)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,83,254,229)(2,84,255,230)(3,85,256,231)(4,86,257,232)(5,87,258,233)(6,88,259,234)(7,89,260,235)(8,90,261,236)(9,91,262,237)(10,92,263,238)(11,93,264,239)(12,94,265,240)(13,95,266,201)(14,96,267,202)(15,97,268,203)(16,98,269,204)(17,99,270,205)(18,100,271,206)(19,101,272,207)(20,102,273,208)(21,103,274,209)(22,104,275,210)(23,105,276,211)(24,106,277,212)(25,107,278,213)(26,108,279,214)(27,109,280,215)(28,110,241,216)(29,111,242,217)(30,112,243,218)(31,113,244,219)(32,114,245,220)(33,115,246,221)(34,116,247,222)(35,117,248,223)(36,118,249,224)(37,119,250,225)(38,120,251,226)(39,81,252,227)(40,82,253,228)(41,145,196,313)(42,146,197,314)(43,147,198,315)(44,148,199,316)(45,149,200,317)(46,150,161,318)(47,151,162,319)(48,152,163,320)(49,153,164,281)(50,154,165,282)(51,155,166,283)(52,156,167,284)(53,157,168,285)(54,158,169,286)(55,159,170,287)(56,160,171,288)(57,121,172,289)(58,122,173,290)(59,123,174,291)(60,124,175,292)(61,125,176,293)(62,126,177,294)(63,127,178,295)(64,128,179,296)(65,129,180,297)(66,130,181,298)(67,131,182,299)(68,132,183,300)(69,133,184,301)(70,134,185,302)(71,135,186,303)(72,136,187,304)(73,137,188,305)(74,138,189,306)(75,139,190,307)(76,140,191,308)(77,141,192,309)(78,142,193,310)(79,143,194,311)(80,144,195,312), (1,179,254,64)(2,180,255,65)(3,181,256,66)(4,182,257,67)(5,183,258,68)(6,184,259,69)(7,185,260,70)(8,186,261,71)(9,187,262,72)(10,188,263,73)(11,189,264,74)(12,190,265,75)(13,191,266,76)(14,192,267,77)(15,193,268,78)(16,194,269,79)(17,195,270,80)(18,196,271,41)(19,197,272,42)(20,198,273,43)(21,199,274,44)(22,200,275,45)(23,161,276,46)(24,162,277,47)(25,163,278,48)(26,164,279,49)(27,165,280,50)(28,166,241,51)(29,167,242,52)(30,168,243,53)(31,169,244,54)(32,170,245,55)(33,171,246,56)(34,172,247,57)(35,173,248,58)(36,174,249,59)(37,175,250,60)(38,176,251,61)(39,177,252,62)(40,178,253,63)(81,126,227,294)(82,127,228,295)(83,128,229,296)(84,129,230,297)(85,130,231,298)(86,131,232,299)(87,132,233,300)(88,133,234,301)(89,134,235,302)(90,135,236,303)(91,136,237,304)(92,137,238,305)(93,138,239,306)(94,139,240,307)(95,140,201,308)(96,141,202,309)(97,142,203,310)(98,143,204,311)(99,144,205,312)(100,145,206,313)(101,146,207,314)(102,147,208,315)(103,148,209,316)(104,149,210,317)(105,150,211,318)(106,151,212,319)(107,152,213,320)(108,153,214,281)(109,154,215,282)(110,155,216,283)(111,156,217,284)(112,157,218,285)(113,158,219,286)(114,159,220,287)(115,160,221,288)(116,121,222,289)(117,122,223,290)(118,123,224,291)(119,124,225,292)(120,125,226,293)>;
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)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,83,254,229)(2,84,255,230)(3,85,256,231)(4,86,257,232)(5,87,258,233)(6,88,259,234)(7,89,260,235)(8,90,261,236)(9,91,262,237)(10,92,263,238)(11,93,264,239)(12,94,265,240)(13,95,266,201)(14,96,267,202)(15,97,268,203)(16,98,269,204)(17,99,270,205)(18,100,271,206)(19,101,272,207)(20,102,273,208)(21,103,274,209)(22,104,275,210)(23,105,276,211)(24,106,277,212)(25,107,278,213)(26,108,279,214)(27,109,280,215)(28,110,241,216)(29,111,242,217)(30,112,243,218)(31,113,244,219)(32,114,245,220)(33,115,246,221)(34,116,247,222)(35,117,248,223)(36,118,249,224)(37,119,250,225)(38,120,251,226)(39,81,252,227)(40,82,253,228)(41,145,196,313)(42,146,197,314)(43,147,198,315)(44,148,199,316)(45,149,200,317)(46,150,161,318)(47,151,162,319)(48,152,163,320)(49,153,164,281)(50,154,165,282)(51,155,166,283)(52,156,167,284)(53,157,168,285)(54,158,169,286)(55,159,170,287)(56,160,171,288)(57,121,172,289)(58,122,173,290)(59,123,174,291)(60,124,175,292)(61,125,176,293)(62,126,177,294)(63,127,178,295)(64,128,179,296)(65,129,180,297)(66,130,181,298)(67,131,182,299)(68,132,183,300)(69,133,184,301)(70,134,185,302)(71,135,186,303)(72,136,187,304)(73,137,188,305)(74,138,189,306)(75,139,190,307)(76,140,191,308)(77,141,192,309)(78,142,193,310)(79,143,194,311)(80,144,195,312), (1,179,254,64)(2,180,255,65)(3,181,256,66)(4,182,257,67)(5,183,258,68)(6,184,259,69)(7,185,260,70)(8,186,261,71)(9,187,262,72)(10,188,263,73)(11,189,264,74)(12,190,265,75)(13,191,266,76)(14,192,267,77)(15,193,268,78)(16,194,269,79)(17,195,270,80)(18,196,271,41)(19,197,272,42)(20,198,273,43)(21,199,274,44)(22,200,275,45)(23,161,276,46)(24,162,277,47)(25,163,278,48)(26,164,279,49)(27,165,280,50)(28,166,241,51)(29,167,242,52)(30,168,243,53)(31,169,244,54)(32,170,245,55)(33,171,246,56)(34,172,247,57)(35,173,248,58)(36,174,249,59)(37,175,250,60)(38,176,251,61)(39,177,252,62)(40,178,253,63)(81,126,227,294)(82,127,228,295)(83,128,229,296)(84,129,230,297)(85,130,231,298)(86,131,232,299)(87,132,233,300)(88,133,234,301)(89,134,235,302)(90,135,236,303)(91,136,237,304)(92,137,238,305)(93,138,239,306)(94,139,240,307)(95,140,201,308)(96,141,202,309)(97,142,203,310)(98,143,204,311)(99,144,205,312)(100,145,206,313)(101,146,207,314)(102,147,208,315)(103,148,209,316)(104,149,210,317)(105,150,211,318)(106,151,212,319)(107,152,213,320)(108,153,214,281)(109,154,215,282)(110,155,216,283)(111,156,217,284)(112,157,218,285)(113,158,219,286)(114,159,220,287)(115,160,221,288)(116,121,222,289)(117,122,223,290)(118,123,224,291)(119,124,225,292)(120,125,226,293) );
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),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,83,254,229),(2,84,255,230),(3,85,256,231),(4,86,257,232),(5,87,258,233),(6,88,259,234),(7,89,260,235),(8,90,261,236),(9,91,262,237),(10,92,263,238),(11,93,264,239),(12,94,265,240),(13,95,266,201),(14,96,267,202),(15,97,268,203),(16,98,269,204),(17,99,270,205),(18,100,271,206),(19,101,272,207),(20,102,273,208),(21,103,274,209),(22,104,275,210),(23,105,276,211),(24,106,277,212),(25,107,278,213),(26,108,279,214),(27,109,280,215),(28,110,241,216),(29,111,242,217),(30,112,243,218),(31,113,244,219),(32,114,245,220),(33,115,246,221),(34,116,247,222),(35,117,248,223),(36,118,249,224),(37,119,250,225),(38,120,251,226),(39,81,252,227),(40,82,253,228),(41,145,196,313),(42,146,197,314),(43,147,198,315),(44,148,199,316),(45,149,200,317),(46,150,161,318),(47,151,162,319),(48,152,163,320),(49,153,164,281),(50,154,165,282),(51,155,166,283),(52,156,167,284),(53,157,168,285),(54,158,169,286),(55,159,170,287),(56,160,171,288),(57,121,172,289),(58,122,173,290),(59,123,174,291),(60,124,175,292),(61,125,176,293),(62,126,177,294),(63,127,178,295),(64,128,179,296),(65,129,180,297),(66,130,181,298),(67,131,182,299),(68,132,183,300),(69,133,184,301),(70,134,185,302),(71,135,186,303),(72,136,187,304),(73,137,188,305),(74,138,189,306),(75,139,190,307),(76,140,191,308),(77,141,192,309),(78,142,193,310),(79,143,194,311),(80,144,195,312)], [(1,179,254,64),(2,180,255,65),(3,181,256,66),(4,182,257,67),(5,183,258,68),(6,184,259,69),(7,185,260,70),(8,186,261,71),(9,187,262,72),(10,188,263,73),(11,189,264,74),(12,190,265,75),(13,191,266,76),(14,192,267,77),(15,193,268,78),(16,194,269,79),(17,195,270,80),(18,196,271,41),(19,197,272,42),(20,198,273,43),(21,199,274,44),(22,200,275,45),(23,161,276,46),(24,162,277,47),(25,163,278,48),(26,164,279,49),(27,165,280,50),(28,166,241,51),(29,167,242,52),(30,168,243,53),(31,169,244,54),(32,170,245,55),(33,171,246,56),(34,172,247,57),(35,173,248,58),(36,174,249,59),(37,175,250,60),(38,176,251,61),(39,177,252,62),(40,178,253,63),(81,126,227,294),(82,127,228,295),(83,128,229,296),(84,129,230,297),(85,130,231,298),(86,131,232,299),(87,132,233,300),(88,133,234,301),(89,134,235,302),(90,135,236,303),(91,136,237,304),(92,137,238,305),(93,138,239,306),(94,139,240,307),(95,140,201,308),(96,141,202,309),(97,142,203,310),(98,143,204,311),(99,144,205,312),(100,145,206,313),(101,146,207,314),(102,147,208,315),(103,148,209,316),(104,149,210,317),(105,150,211,318),(106,151,212,319),(107,152,213,320),(108,153,214,281),(109,154,215,282),(110,155,216,283),(111,156,217,284),(112,157,218,285),(113,158,219,286),(114,159,220,287),(115,160,221,288),(116,121,222,289),(117,122,223,290),(118,123,224,291),(119,124,225,292),(120,125,226,293)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20BL | 40A | ··· | 40AF | 40AG | ··· | 40CB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 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 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | Q8 | C4○D4 | C8○D4 | C5×Q8 | C5×C4○D4 | C5×C8○D4 |
| kernel | Q8×C40 | C4×C40 | C5×C4⋊C8 | Q8×C20 | C5×C4⋊C4 | Q8×C10 | C8×Q8 | C5×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C40 | C20 | C10 | C8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 16 | 12 | 12 | 4 | 24 | 8 | 64 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of Q8×C40 ►in GL3(𝔽41) generated by
| 27 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 40 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 40 | 0 | 0 |
| 0 | 15 | 26 |
| 0 | 26 | 26 |
G:=sub<GL(3,GF(41))| [27,0,0,0,23,0,0,0,23],[40,0,0,0,0,40,0,1,0],[40,0,0,0,15,26,0,26,26] >;
Q8×C40 in GAP, Magma, Sage, TeX
Q_8\times C_{40} % in TeX
G:=Group("Q8xC40"); // GroupNames label
G:=SmallGroup(320,946);
// by ID
G=gap.SmallGroup(320,946);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,646,124]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations