metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6Q8, C8⋊6Dic10, C42.12D10, Dic5.4M4(2), C5⋊5(C8⋊4Q8), C8⋊C4.5D5, C10.16(C4×Q8), C20.81(C2×Q8), C20⋊3C8.5C2, (C2×C8).154D10, C4⋊Dic5.20C4, C40⋊8C4.12C2, C2.9(C4×Dic10), C2.8(D5×M4(2)), C10.44(C8○D4), (C4×C20).12C22, (C4×Dic10).4C2, (C8×Dic5).27C2, C4.46(C2×Dic10), C4.128(C4○D20), C20.244(C4○D4), (C2×C40).224C22, (C2×C20).809C23, (C2×Dic10).19C4, C10.D4.14C4, C10.48(C2×M4(2)), C20.8Q8.15C2, C2.6(D20.2C4), (C4×Dic5).200C22, (C2×C4).28(C4×D5), (C5×C8⋊C4).4C2, C22.97(C2×C4×D5), (C2×C20).209(C2×C4), (C2×Dic5).15(C2×C4), (C2×C4).751(C22×D5), (C2×C10).165(C22×C4), (C2×C5⋊2C8).301C22, SmallGroup(320,328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊Q8
G = < a,b,c | a40=b4=1, c2=b2, bab-1=a21, cac-1=a9, cbc-1=b-1 >
Subgroups: 254 in 94 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, C8⋊C4, C8⋊C4, C4⋊C8, C4×Q8, C5⋊2C8, C40, C40, Dic10, C2×Dic5, C2×C20, C8⋊4Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C20⋊3C8, C8×Dic5, C20.8Q8, C40⋊8C4, C5×C8⋊C4, C4×Dic10, C40⋊Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, M4(2), C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C2×M4(2), C8○D4, Dic10, C4×D5, C22×D5, C8⋊4Q8, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10, D5×M4(2), D20.2C4, C40⋊Q8
►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 87 253 232)(2 108 254 213)(3 89 255 234)(4 110 256 215)(5 91 257 236)(6 112 258 217)(7 93 259 238)(8 114 260 219)(9 95 261 240)(10 116 262 221)(11 97 263 202)(12 118 264 223)(13 99 265 204)(14 120 266 225)(15 101 267 206)(16 82 268 227)(17 103 269 208)(18 84 270 229)(19 105 271 210)(20 86 272 231)(21 107 273 212)(22 88 274 233)(23 109 275 214)(24 90 276 235)(25 111 277 216)(26 92 278 237)(27 113 279 218)(28 94 280 239)(29 115 241 220)(30 96 242 201)(31 117 243 222)(32 98 244 203)(33 119 245 224)(34 100 246 205)(35 81 247 226)(36 102 248 207)(37 83 249 228)(38 104 250 209)(39 85 251 230)(40 106 252 211)(41 121 183 286)(42 142 184 307)(43 123 185 288)(44 144 186 309)(45 125 187 290)(46 146 188 311)(47 127 189 292)(48 148 190 313)(49 129 191 294)(50 150 192 315)(51 131 193 296)(52 152 194 317)(53 133 195 298)(54 154 196 319)(55 135 197 300)(56 156 198 281)(57 137 199 302)(58 158 200 283)(59 139 161 304)(60 160 162 285)(61 141 163 306)(62 122 164 287)(63 143 165 308)(64 124 166 289)(65 145 167 310)(66 126 168 291)(67 147 169 312)(68 128 170 293)(69 149 171 314)(70 130 172 295)(71 151 173 316)(72 132 174 297)(73 153 175 318)(74 134 176 299)(75 155 177 320)(76 136 178 301)(77 157 179 282)(78 138 180 303)(79 159 181 284)(80 140 182 305)
(1 301 253 136)(2 310 254 145)(3 319 255 154)(4 288 256 123)(5 297 257 132)(6 306 258 141)(7 315 259 150)(8 284 260 159)(9 293 261 128)(10 302 262 137)(11 311 263 146)(12 320 264 155)(13 289 265 124)(14 298 266 133)(15 307 267 142)(16 316 268 151)(17 285 269 160)(18 294 270 129)(19 303 271 138)(20 312 272 147)(21 281 273 156)(22 290 274 125)(23 299 275 134)(24 308 276 143)(25 317 277 152)(26 286 278 121)(27 295 279 130)(28 304 280 139)(29 313 241 148)(30 282 242 157)(31 291 243 126)(32 300 244 135)(33 309 245 144)(34 318 246 153)(35 287 247 122)(36 296 248 131)(37 305 249 140)(38 314 250 149)(39 283 251 158)(40 292 252 127)(41 92 183 237)(42 101 184 206)(43 110 185 215)(44 119 186 224)(45 88 187 233)(46 97 188 202)(47 106 189 211)(48 115 190 220)(49 84 191 229)(50 93 192 238)(51 102 193 207)(52 111 194 216)(53 120 195 225)(54 89 196 234)(55 98 197 203)(56 107 198 212)(57 116 199 221)(58 85 200 230)(59 94 161 239)(60 103 162 208)(61 112 163 217)(62 81 164 226)(63 90 165 235)(64 99 166 204)(65 108 167 213)(66 117 168 222)(67 86 169 231)(68 95 170 240)(69 104 171 209)(70 113 172 218)(71 82 173 227)(72 91 174 236)(73 100 175 205)(74 109 176 214)(75 118 177 223)(76 87 178 232)(77 96 179 201)(78 105 180 210)(79 114 181 219)(80 83 182 228)
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), (1,87,253,232)(2,108,254,213)(3,89,255,234)(4,110,256,215)(5,91,257,236)(6,112,258,217)(7,93,259,238)(8,114,260,219)(9,95,261,240)(10,116,262,221)(11,97,263,202)(12,118,264,223)(13,99,265,204)(14,120,266,225)(15,101,267,206)(16,82,268,227)(17,103,269,208)(18,84,270,229)(19,105,271,210)(20,86,272,231)(21,107,273,212)(22,88,274,233)(23,109,275,214)(24,90,276,235)(25,111,277,216)(26,92,278,237)(27,113,279,218)(28,94,280,239)(29,115,241,220)(30,96,242,201)(31,117,243,222)(32,98,244,203)(33,119,245,224)(34,100,246,205)(35,81,247,226)(36,102,248,207)(37,83,249,228)(38,104,250,209)(39,85,251,230)(40,106,252,211)(41,121,183,286)(42,142,184,307)(43,123,185,288)(44,144,186,309)(45,125,187,290)(46,146,188,311)(47,127,189,292)(48,148,190,313)(49,129,191,294)(50,150,192,315)(51,131,193,296)(52,152,194,317)(53,133,195,298)(54,154,196,319)(55,135,197,300)(56,156,198,281)(57,137,199,302)(58,158,200,283)(59,139,161,304)(60,160,162,285)(61,141,163,306)(62,122,164,287)(63,143,165,308)(64,124,166,289)(65,145,167,310)(66,126,168,291)(67,147,169,312)(68,128,170,293)(69,149,171,314)(70,130,172,295)(71,151,173,316)(72,132,174,297)(73,153,175,318)(74,134,176,299)(75,155,177,320)(76,136,178,301)(77,157,179,282)(78,138,180,303)(79,159,181,284)(80,140,182,305), (1,301,253,136)(2,310,254,145)(3,319,255,154)(4,288,256,123)(5,297,257,132)(6,306,258,141)(7,315,259,150)(8,284,260,159)(9,293,261,128)(10,302,262,137)(11,311,263,146)(12,320,264,155)(13,289,265,124)(14,298,266,133)(15,307,267,142)(16,316,268,151)(17,285,269,160)(18,294,270,129)(19,303,271,138)(20,312,272,147)(21,281,273,156)(22,290,274,125)(23,299,275,134)(24,308,276,143)(25,317,277,152)(26,286,278,121)(27,295,279,130)(28,304,280,139)(29,313,241,148)(30,282,242,157)(31,291,243,126)(32,300,244,135)(33,309,245,144)(34,318,246,153)(35,287,247,122)(36,296,248,131)(37,305,249,140)(38,314,250,149)(39,283,251,158)(40,292,252,127)(41,92,183,237)(42,101,184,206)(43,110,185,215)(44,119,186,224)(45,88,187,233)(46,97,188,202)(47,106,189,211)(48,115,190,220)(49,84,191,229)(50,93,192,238)(51,102,193,207)(52,111,194,216)(53,120,195,225)(54,89,196,234)(55,98,197,203)(56,107,198,212)(57,116,199,221)(58,85,200,230)(59,94,161,239)(60,103,162,208)(61,112,163,217)(62,81,164,226)(63,90,165,235)(64,99,166,204)(65,108,167,213)(66,117,168,222)(67,86,169,231)(68,95,170,240)(69,104,171,209)(70,113,172,218)(71,82,173,227)(72,91,174,236)(73,100,175,205)(74,109,176,214)(75,118,177,223)(76,87,178,232)(77,96,179,201)(78,105,180,210)(79,114,181,219)(80,83,182,228)>;
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), (1,87,253,232)(2,108,254,213)(3,89,255,234)(4,110,256,215)(5,91,257,236)(6,112,258,217)(7,93,259,238)(8,114,260,219)(9,95,261,240)(10,116,262,221)(11,97,263,202)(12,118,264,223)(13,99,265,204)(14,120,266,225)(15,101,267,206)(16,82,268,227)(17,103,269,208)(18,84,270,229)(19,105,271,210)(20,86,272,231)(21,107,273,212)(22,88,274,233)(23,109,275,214)(24,90,276,235)(25,111,277,216)(26,92,278,237)(27,113,279,218)(28,94,280,239)(29,115,241,220)(30,96,242,201)(31,117,243,222)(32,98,244,203)(33,119,245,224)(34,100,246,205)(35,81,247,226)(36,102,248,207)(37,83,249,228)(38,104,250,209)(39,85,251,230)(40,106,252,211)(41,121,183,286)(42,142,184,307)(43,123,185,288)(44,144,186,309)(45,125,187,290)(46,146,188,311)(47,127,189,292)(48,148,190,313)(49,129,191,294)(50,150,192,315)(51,131,193,296)(52,152,194,317)(53,133,195,298)(54,154,196,319)(55,135,197,300)(56,156,198,281)(57,137,199,302)(58,158,200,283)(59,139,161,304)(60,160,162,285)(61,141,163,306)(62,122,164,287)(63,143,165,308)(64,124,166,289)(65,145,167,310)(66,126,168,291)(67,147,169,312)(68,128,170,293)(69,149,171,314)(70,130,172,295)(71,151,173,316)(72,132,174,297)(73,153,175,318)(74,134,176,299)(75,155,177,320)(76,136,178,301)(77,157,179,282)(78,138,180,303)(79,159,181,284)(80,140,182,305), (1,301,253,136)(2,310,254,145)(3,319,255,154)(4,288,256,123)(5,297,257,132)(6,306,258,141)(7,315,259,150)(8,284,260,159)(9,293,261,128)(10,302,262,137)(11,311,263,146)(12,320,264,155)(13,289,265,124)(14,298,266,133)(15,307,267,142)(16,316,268,151)(17,285,269,160)(18,294,270,129)(19,303,271,138)(20,312,272,147)(21,281,273,156)(22,290,274,125)(23,299,275,134)(24,308,276,143)(25,317,277,152)(26,286,278,121)(27,295,279,130)(28,304,280,139)(29,313,241,148)(30,282,242,157)(31,291,243,126)(32,300,244,135)(33,309,245,144)(34,318,246,153)(35,287,247,122)(36,296,248,131)(37,305,249,140)(38,314,250,149)(39,283,251,158)(40,292,252,127)(41,92,183,237)(42,101,184,206)(43,110,185,215)(44,119,186,224)(45,88,187,233)(46,97,188,202)(47,106,189,211)(48,115,190,220)(49,84,191,229)(50,93,192,238)(51,102,193,207)(52,111,194,216)(53,120,195,225)(54,89,196,234)(55,98,197,203)(56,107,198,212)(57,116,199,221)(58,85,200,230)(59,94,161,239)(60,103,162,208)(61,112,163,217)(62,81,164,226)(63,90,165,235)(64,99,166,204)(65,108,167,213)(66,117,168,222)(67,86,169,231)(68,95,170,240)(69,104,171,209)(70,113,172,218)(71,82,173,227)(72,91,174,236)(73,100,175,205)(74,109,176,214)(75,118,177,223)(76,87,178,232)(77,96,179,201)(78,105,180,210)(79,114,181,219)(80,83,182,228) );
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)], [(1,87,253,232),(2,108,254,213),(3,89,255,234),(4,110,256,215),(5,91,257,236),(6,112,258,217),(7,93,259,238),(8,114,260,219),(9,95,261,240),(10,116,262,221),(11,97,263,202),(12,118,264,223),(13,99,265,204),(14,120,266,225),(15,101,267,206),(16,82,268,227),(17,103,269,208),(18,84,270,229),(19,105,271,210),(20,86,272,231),(21,107,273,212),(22,88,274,233),(23,109,275,214),(24,90,276,235),(25,111,277,216),(26,92,278,237),(27,113,279,218),(28,94,280,239),(29,115,241,220),(30,96,242,201),(31,117,243,222),(32,98,244,203),(33,119,245,224),(34,100,246,205),(35,81,247,226),(36,102,248,207),(37,83,249,228),(38,104,250,209),(39,85,251,230),(40,106,252,211),(41,121,183,286),(42,142,184,307),(43,123,185,288),(44,144,186,309),(45,125,187,290),(46,146,188,311),(47,127,189,292),(48,148,190,313),(49,129,191,294),(50,150,192,315),(51,131,193,296),(52,152,194,317),(53,133,195,298),(54,154,196,319),(55,135,197,300),(56,156,198,281),(57,137,199,302),(58,158,200,283),(59,139,161,304),(60,160,162,285),(61,141,163,306),(62,122,164,287),(63,143,165,308),(64,124,166,289),(65,145,167,310),(66,126,168,291),(67,147,169,312),(68,128,170,293),(69,149,171,314),(70,130,172,295),(71,151,173,316),(72,132,174,297),(73,153,175,318),(74,134,176,299),(75,155,177,320),(76,136,178,301),(77,157,179,282),(78,138,180,303),(79,159,181,284),(80,140,182,305)], [(1,301,253,136),(2,310,254,145),(3,319,255,154),(4,288,256,123),(5,297,257,132),(6,306,258,141),(7,315,259,150),(8,284,260,159),(9,293,261,128),(10,302,262,137),(11,311,263,146),(12,320,264,155),(13,289,265,124),(14,298,266,133),(15,307,267,142),(16,316,268,151),(17,285,269,160),(18,294,270,129),(19,303,271,138),(20,312,272,147),(21,281,273,156),(22,290,274,125),(23,299,275,134),(24,308,276,143),(25,317,277,152),(26,286,278,121),(27,295,279,130),(28,304,280,139),(29,313,241,148),(30,282,242,157),(31,291,243,126),(32,300,244,135),(33,309,245,144),(34,318,246,153),(35,287,247,122),(36,296,248,131),(37,305,249,140),(38,314,250,149),(39,283,251,158),(40,292,252,127),(41,92,183,237),(42,101,184,206),(43,110,185,215),(44,119,186,224),(45,88,187,233),(46,97,188,202),(47,106,189,211),(48,115,190,220),(49,84,191,229),(50,93,192,238),(51,102,193,207),(52,111,194,216),(53,120,195,225),(54,89,196,234),(55,98,197,203),(56,107,198,212),(57,116,199,221),(58,85,200,230),(59,94,161,239),(60,103,162,208),(61,112,163,217),(62,81,164,226),(63,90,165,235),(64,99,166,204),(65,108,167,213),(66,117,168,222),(67,86,169,231),(68,95,170,240),(69,104,171,209),(70,113,172,218),(71,82,173,227),(72,91,174,236),(73,100,175,205),(74,109,176,214),(75,118,177,223),(76,87,178,232),(77,96,179,201),(78,105,180,210),(79,114,181,219),(80,83,182,228)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | Dic10 | C4×D5 | C4○D20 | D5×M4(2) | D20.2C4 |
| kernel | C40⋊Q8 | C20⋊3C8 | C8×Dic5 | C20.8Q8 | C40⋊8C4 | C5×C8⋊C4 | C4×Dic10 | C10.D4 | C4⋊Dic5 | C2×Dic10 | C40 | C8⋊C4 | Dic5 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C40⋊Q8 ►in GL6(𝔽41)
| 32 | 21 | 0 | 0 | 0 | 0 |
| 25 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 19 |
| 0 | 0 | 0 | 0 | 22 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 37 | 0 | 0 |
| 0 | 0 | 6 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 22 | 0 | 0 |
| 0 | 0 | 2 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 20 |
| 0 | 0 | 0 | 0 | 38 | 15 |
G:=sub<GL(6,GF(41))| [32,25,0,0,0,0,21,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,22,22,0,0,0,0,19,32],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,8,6,0,0,0,0,37,33,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,2,0,0,0,0,22,18,0,0,0,0,0,0,26,38,0,0,0,0,20,15] >;
C40⋊Q8 in GAP, Magma, Sage, TeX
C_{40}\rtimes Q_8 % in TeX
G:=Group("C40:Q8"); // GroupNames label
G:=SmallGroup(320,328);
// by ID
G=gap.SmallGroup(320,328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,b*a*b^-1=a^21,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations