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G = C404Q8order 320 = 26·5

4th semidirect product of C40 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C404Q8, C83Dic10, C53(C8⋊Q8), C52C82Q8, C4.27(Q8×D5), C4⋊C4.44D10, (C2×C8).64D10, C2.D8.8D5, C20⋊Q8.10C2, C20.59(C2×Q8), C408C4.3C2, C2.12(C20⋊Q8), C10.17(C4⋊Q8), C406C4.10C2, (C2×Dic5).54D4, C4.24(C2×Dic10), C10.D8.9C2, C22.225(D4×D5), C20.Q8.7C2, C4.Dic10.8C2, C2.21(D8⋊D5), C10.39(C8⋊C22), (C2×C20).292C23, (C2×C40).142C22, C2.20(Q16⋊D5), C10.67(C8.C22), C4⋊Dic5.118C22, (C4×Dic5).41C22, (C5×C2.D8).7C2, (C2×C10).297(C2×D4), (C5×C4⋊C4).85C22, (C2×C52C8).66C22, (C2×C4).395(C22×D5), SmallGroup(320,503)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C404Q8
C1C5C10C2×C10C2×C20C4×Dic5C408C4 — C404Q8
C5C10C2×C20 — C404Q8
C1C22C2×C4C2.D8

Generators and relations for C404Q8
 G = < a,b,c | a40=b4=1, c2=b2, bab-1=a31, cac-1=a29, cbc-1=b-1 >

Subgroups: 334 in 90 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×6], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×5], C2×C8, C2×C8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C8⋊C4, C4.Q8 [×2], C2.D8, C2.D8, C42.C2, C4⋊Q8, C52C8 [×2], C40 [×2], Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C8⋊Q8, C2×C52C8, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C10.D8, C20.Q8, C408C4, C406C4, C5×C2.D8, C20⋊Q8, C4.Dic10, C404Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C8⋊C22, C8.C22, Dic10 [×2], C22×D5, C8⋊Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, D8⋊D5, Q16⋊D5, C404Q8

Smallest permutation representation of C404Q8
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 63 281 126)(2 54 282 157)(3 45 283 148)(4 76 284 139)(5 67 285 130)(6 58 286 121)(7 49 287 152)(8 80 288 143)(9 71 289 134)(10 62 290 125)(11 53 291 156)(12 44 292 147)(13 75 293 138)(14 66 294 129)(15 57 295 160)(16 48 296 151)(17 79 297 142)(18 70 298 133)(19 61 299 124)(20 52 300 155)(21 43 301 146)(22 74 302 137)(23 65 303 128)(24 56 304 159)(25 47 305 150)(26 78 306 141)(27 69 307 132)(28 60 308 123)(29 51 309 154)(30 42 310 145)(31 73 311 136)(32 64 312 127)(33 55 313 158)(34 46 314 149)(35 77 315 140)(36 68 316 131)(37 59 317 122)(38 50 318 153)(39 41 319 144)(40 72 320 135)(81 256 238 185)(82 247 239 176)(83 278 240 167)(84 269 201 198)(85 260 202 189)(86 251 203 180)(87 242 204 171)(88 273 205 162)(89 264 206 193)(90 255 207 184)(91 246 208 175)(92 277 209 166)(93 268 210 197)(94 259 211 188)(95 250 212 179)(96 241 213 170)(97 272 214 161)(98 263 215 192)(99 254 216 183)(100 245 217 174)(101 276 218 165)(102 267 219 196)(103 258 220 187)(104 249 221 178)(105 280 222 169)(106 271 223 200)(107 262 224 191)(108 253 225 182)(109 244 226 173)(110 275 227 164)(111 266 228 195)(112 257 229 186)(113 248 230 177)(114 279 231 168)(115 270 232 199)(116 261 233 190)(117 252 234 181)(118 243 235 172)(119 274 236 163)(120 265 237 194)
(1 179 281 250)(2 168 282 279)(3 197 283 268)(4 186 284 257)(5 175 285 246)(6 164 286 275)(7 193 287 264)(8 182 288 253)(9 171 289 242)(10 200 290 271)(11 189 291 260)(12 178 292 249)(13 167 293 278)(14 196 294 267)(15 185 295 256)(16 174 296 245)(17 163 297 274)(18 192 298 263)(19 181 299 252)(20 170 300 241)(21 199 301 270)(22 188 302 259)(23 177 303 248)(24 166 304 277)(25 195 305 266)(26 184 306 255)(27 173 307 244)(28 162 308 273)(29 191 309 262)(30 180 310 251)(31 169 311 280)(32 198 312 269)(33 187 313 258)(34 176 314 247)(35 165 315 276)(36 194 316 265)(37 183 317 254)(38 172 318 243)(39 161 319 272)(40 190 320 261)(41 214 144 97)(42 203 145 86)(43 232 146 115)(44 221 147 104)(45 210 148 93)(46 239 149 82)(47 228 150 111)(48 217 151 100)(49 206 152 89)(50 235 153 118)(51 224 154 107)(52 213 155 96)(53 202 156 85)(54 231 157 114)(55 220 158 103)(56 209 159 92)(57 238 160 81)(58 227 121 110)(59 216 122 99)(60 205 123 88)(61 234 124 117)(62 223 125 106)(63 212 126 95)(64 201 127 84)(65 230 128 113)(66 219 129 102)(67 208 130 91)(68 237 131 120)(69 226 132 109)(70 215 133 98)(71 204 134 87)(72 233 135 116)(73 222 136 105)(74 211 137 94)(75 240 138 83)(76 229 139 112)(77 218 140 101)(78 207 141 90)(79 236 142 119)(80 225 143 108)

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), (1,63,281,126)(2,54,282,157)(3,45,283,148)(4,76,284,139)(5,67,285,130)(6,58,286,121)(7,49,287,152)(8,80,288,143)(9,71,289,134)(10,62,290,125)(11,53,291,156)(12,44,292,147)(13,75,293,138)(14,66,294,129)(15,57,295,160)(16,48,296,151)(17,79,297,142)(18,70,298,133)(19,61,299,124)(20,52,300,155)(21,43,301,146)(22,74,302,137)(23,65,303,128)(24,56,304,159)(25,47,305,150)(26,78,306,141)(27,69,307,132)(28,60,308,123)(29,51,309,154)(30,42,310,145)(31,73,311,136)(32,64,312,127)(33,55,313,158)(34,46,314,149)(35,77,315,140)(36,68,316,131)(37,59,317,122)(38,50,318,153)(39,41,319,144)(40,72,320,135)(81,256,238,185)(82,247,239,176)(83,278,240,167)(84,269,201,198)(85,260,202,189)(86,251,203,180)(87,242,204,171)(88,273,205,162)(89,264,206,193)(90,255,207,184)(91,246,208,175)(92,277,209,166)(93,268,210,197)(94,259,211,188)(95,250,212,179)(96,241,213,170)(97,272,214,161)(98,263,215,192)(99,254,216,183)(100,245,217,174)(101,276,218,165)(102,267,219,196)(103,258,220,187)(104,249,221,178)(105,280,222,169)(106,271,223,200)(107,262,224,191)(108,253,225,182)(109,244,226,173)(110,275,227,164)(111,266,228,195)(112,257,229,186)(113,248,230,177)(114,279,231,168)(115,270,232,199)(116,261,233,190)(117,252,234,181)(118,243,235,172)(119,274,236,163)(120,265,237,194), (1,179,281,250)(2,168,282,279)(3,197,283,268)(4,186,284,257)(5,175,285,246)(6,164,286,275)(7,193,287,264)(8,182,288,253)(9,171,289,242)(10,200,290,271)(11,189,291,260)(12,178,292,249)(13,167,293,278)(14,196,294,267)(15,185,295,256)(16,174,296,245)(17,163,297,274)(18,192,298,263)(19,181,299,252)(20,170,300,241)(21,199,301,270)(22,188,302,259)(23,177,303,248)(24,166,304,277)(25,195,305,266)(26,184,306,255)(27,173,307,244)(28,162,308,273)(29,191,309,262)(30,180,310,251)(31,169,311,280)(32,198,312,269)(33,187,313,258)(34,176,314,247)(35,165,315,276)(36,194,316,265)(37,183,317,254)(38,172,318,243)(39,161,319,272)(40,190,320,261)(41,214,144,97)(42,203,145,86)(43,232,146,115)(44,221,147,104)(45,210,148,93)(46,239,149,82)(47,228,150,111)(48,217,151,100)(49,206,152,89)(50,235,153,118)(51,224,154,107)(52,213,155,96)(53,202,156,85)(54,231,157,114)(55,220,158,103)(56,209,159,92)(57,238,160,81)(58,227,121,110)(59,216,122,99)(60,205,123,88)(61,234,124,117)(62,223,125,106)(63,212,126,95)(64,201,127,84)(65,230,128,113)(66,219,129,102)(67,208,130,91)(68,237,131,120)(69,226,132,109)(70,215,133,98)(71,204,134,87)(72,233,135,116)(73,222,136,105)(74,211,137,94)(75,240,138,83)(76,229,139,112)(77,218,140,101)(78,207,141,90)(79,236,142,119)(80,225,143,108)>;

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), (1,63,281,126)(2,54,282,157)(3,45,283,148)(4,76,284,139)(5,67,285,130)(6,58,286,121)(7,49,287,152)(8,80,288,143)(9,71,289,134)(10,62,290,125)(11,53,291,156)(12,44,292,147)(13,75,293,138)(14,66,294,129)(15,57,295,160)(16,48,296,151)(17,79,297,142)(18,70,298,133)(19,61,299,124)(20,52,300,155)(21,43,301,146)(22,74,302,137)(23,65,303,128)(24,56,304,159)(25,47,305,150)(26,78,306,141)(27,69,307,132)(28,60,308,123)(29,51,309,154)(30,42,310,145)(31,73,311,136)(32,64,312,127)(33,55,313,158)(34,46,314,149)(35,77,315,140)(36,68,316,131)(37,59,317,122)(38,50,318,153)(39,41,319,144)(40,72,320,135)(81,256,238,185)(82,247,239,176)(83,278,240,167)(84,269,201,198)(85,260,202,189)(86,251,203,180)(87,242,204,171)(88,273,205,162)(89,264,206,193)(90,255,207,184)(91,246,208,175)(92,277,209,166)(93,268,210,197)(94,259,211,188)(95,250,212,179)(96,241,213,170)(97,272,214,161)(98,263,215,192)(99,254,216,183)(100,245,217,174)(101,276,218,165)(102,267,219,196)(103,258,220,187)(104,249,221,178)(105,280,222,169)(106,271,223,200)(107,262,224,191)(108,253,225,182)(109,244,226,173)(110,275,227,164)(111,266,228,195)(112,257,229,186)(113,248,230,177)(114,279,231,168)(115,270,232,199)(116,261,233,190)(117,252,234,181)(118,243,235,172)(119,274,236,163)(120,265,237,194), (1,179,281,250)(2,168,282,279)(3,197,283,268)(4,186,284,257)(5,175,285,246)(6,164,286,275)(7,193,287,264)(8,182,288,253)(9,171,289,242)(10,200,290,271)(11,189,291,260)(12,178,292,249)(13,167,293,278)(14,196,294,267)(15,185,295,256)(16,174,296,245)(17,163,297,274)(18,192,298,263)(19,181,299,252)(20,170,300,241)(21,199,301,270)(22,188,302,259)(23,177,303,248)(24,166,304,277)(25,195,305,266)(26,184,306,255)(27,173,307,244)(28,162,308,273)(29,191,309,262)(30,180,310,251)(31,169,311,280)(32,198,312,269)(33,187,313,258)(34,176,314,247)(35,165,315,276)(36,194,316,265)(37,183,317,254)(38,172,318,243)(39,161,319,272)(40,190,320,261)(41,214,144,97)(42,203,145,86)(43,232,146,115)(44,221,147,104)(45,210,148,93)(46,239,149,82)(47,228,150,111)(48,217,151,100)(49,206,152,89)(50,235,153,118)(51,224,154,107)(52,213,155,96)(53,202,156,85)(54,231,157,114)(55,220,158,103)(56,209,159,92)(57,238,160,81)(58,227,121,110)(59,216,122,99)(60,205,123,88)(61,234,124,117)(62,223,125,106)(63,212,126,95)(64,201,127,84)(65,230,128,113)(66,219,129,102)(67,208,130,91)(68,237,131,120)(69,226,132,109)(70,215,133,98)(71,204,134,87)(72,233,135,116)(73,222,136,105)(74,211,137,94)(75,240,138,83)(76,229,139,112)(77,218,140,101)(78,207,141,90)(79,236,142,119)(80,225,143,108) );

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)], [(1,63,281,126),(2,54,282,157),(3,45,283,148),(4,76,284,139),(5,67,285,130),(6,58,286,121),(7,49,287,152),(8,80,288,143),(9,71,289,134),(10,62,290,125),(11,53,291,156),(12,44,292,147),(13,75,293,138),(14,66,294,129),(15,57,295,160),(16,48,296,151),(17,79,297,142),(18,70,298,133),(19,61,299,124),(20,52,300,155),(21,43,301,146),(22,74,302,137),(23,65,303,128),(24,56,304,159),(25,47,305,150),(26,78,306,141),(27,69,307,132),(28,60,308,123),(29,51,309,154),(30,42,310,145),(31,73,311,136),(32,64,312,127),(33,55,313,158),(34,46,314,149),(35,77,315,140),(36,68,316,131),(37,59,317,122),(38,50,318,153),(39,41,319,144),(40,72,320,135),(81,256,238,185),(82,247,239,176),(83,278,240,167),(84,269,201,198),(85,260,202,189),(86,251,203,180),(87,242,204,171),(88,273,205,162),(89,264,206,193),(90,255,207,184),(91,246,208,175),(92,277,209,166),(93,268,210,197),(94,259,211,188),(95,250,212,179),(96,241,213,170),(97,272,214,161),(98,263,215,192),(99,254,216,183),(100,245,217,174),(101,276,218,165),(102,267,219,196),(103,258,220,187),(104,249,221,178),(105,280,222,169),(106,271,223,200),(107,262,224,191),(108,253,225,182),(109,244,226,173),(110,275,227,164),(111,266,228,195),(112,257,229,186),(113,248,230,177),(114,279,231,168),(115,270,232,199),(116,261,233,190),(117,252,234,181),(118,243,235,172),(119,274,236,163),(120,265,237,194)], [(1,179,281,250),(2,168,282,279),(3,197,283,268),(4,186,284,257),(5,175,285,246),(6,164,286,275),(7,193,287,264),(8,182,288,253),(9,171,289,242),(10,200,290,271),(11,189,291,260),(12,178,292,249),(13,167,293,278),(14,196,294,267),(15,185,295,256),(16,174,296,245),(17,163,297,274),(18,192,298,263),(19,181,299,252),(20,170,300,241),(21,199,301,270),(22,188,302,259),(23,177,303,248),(24,166,304,277),(25,195,305,266),(26,184,306,255),(27,173,307,244),(28,162,308,273),(29,191,309,262),(30,180,310,251),(31,169,311,280),(32,198,312,269),(33,187,313,258),(34,176,314,247),(35,165,315,276),(36,194,316,265),(37,183,317,254),(38,172,318,243),(39,161,319,272),(40,190,320,261),(41,214,144,97),(42,203,145,86),(43,232,146,115),(44,221,147,104),(45,210,148,93),(46,239,149,82),(47,228,150,111),(48,217,151,100),(49,206,152,89),(50,235,153,118),(51,224,154,107),(52,213,155,96),(53,202,156,85),(54,231,157,114),(55,220,158,103),(56,209,159,92),(57,238,160,81),(58,227,121,110),(59,216,122,99),(60,205,123,88),(61,234,124,117),(62,223,125,106),(63,212,126,95),(64,201,127,84),(65,230,128,113),(66,219,129,102),(67,208,130,91),(68,237,131,120),(69,226,132,109),(70,215,133,98),(71,204,134,87),(72,233,135,116),(73,222,136,105),(74,211,137,94),(75,240,138,83),(76,229,139,112),(77,218,140,101),(78,207,141,90),(79,236,142,119),(80,225,143,108)])

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444455888810···102020202020···2040···40
size1111228820204040224420202···244448···84···4

44 irreducible representations

dim111111112222222444444
type++++++++--++++-+--+
imageC1C2C2C2C2C2C2C2Q8Q8D4D5D10D10Dic10C8⋊C22C8.C22Q8×D5D4×D5D8⋊D5Q16⋊D5
kernelC404Q8C10.D8C20.Q8C408C4C406C4C5×C2.D8C20⋊Q8C4.Dic10C52C8C40C2×Dic5C2.D8C4⋊C4C2×C8C8C10C10C4C22C2C2
# reps111111112222428112244

Matrix representation of C404Q8 in GL6(𝔽41)

100000
010000
00116116
00356356
003035116
00635356
,
2590000
17160000
00515283
002613385
002833626
003851528
,
25220000
20160000
0022352115
003191820
0020262235
002321319

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,35,30,6,0,0,6,6,35,35,0,0,11,35,11,35,0,0,6,6,6,6],[25,17,0,0,0,0,9,16,0,0,0,0,0,0,5,26,28,38,0,0,15,13,3,5,0,0,28,38,36,15,0,0,3,5,26,28],[25,20,0,0,0,0,22,16,0,0,0,0,0,0,22,3,20,23,0,0,35,19,26,21,0,0,21,18,22,3,0,0,15,20,35,19] >;

C404Q8 in GAP, Magma, Sage, TeX

C_{40}\rtimes_4Q_8
% in TeX

G:=Group("C40:4Q8");
// GroupNames label

G:=SmallGroup(320,503);
// by ID

G=gap.SmallGroup(320,503);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,254,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=b^-1>;
// generators/relations

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