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G = C402Q8order 320 = 26·5

2nd semidirect product of C40 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C402Q8, C84Dic10, Dic5.6D8, Dic5.4Q16, C52C86Q8, C2.12(D5×D8), C52(C82Q8), C20⋊Q8.8C2, C4.26(Q8×D5), C4⋊C4.43D10, C10.27(C2×D8), C2.D8.4D5, C20.17(C2×Q8), C2.12(D5×Q16), (C2×C8).227D10, C2.11(C20⋊Q8), C10.16(C4⋊Q8), C10.21(C2×Q16), (C8×Dic5).2C2, C405C4.14C2, (C2×C40).79C22, C4.23(C2×Dic10), C10.D8.7C2, C22.223(D4×D5), (C2×C20).290C23, (C2×Dic5).145D4, C4⋊Dic5.116C22, (C4×Dic5).264C22, (C5×C2.D8).5C2, (C2×C10).295(C2×D4), (C5×C4⋊C4).83C22, (C2×C4).393(C22×D5), (C2×C52C8).239C22, SmallGroup(320,501)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C402Q8
C1C5C10C2×C10C2×C20C4×Dic5C8×Dic5 — C402Q8
C5C10C2×C20 — C402Q8
C1C22C2×C4C2.D8

Generators and relations for C402Q8
 G = < a,b,c | a40=b4=1, c2=b2, bab-1=a31, cac-1=a9, cbc-1=b-1 >

Subgroups: 382 in 98 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8, C2×Q8 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C4×C8, C2.D8, C2.D8 [×3], C4⋊Q8 [×2], C52C8 [×2], C40 [×2], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C82Q8, C2×C52C8, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10 [×2], C10.D8 [×2], C8×Dic5, C405C4, C5×C2.D8, C20⋊Q8 [×2], C402Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, D8 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C2×D8, C2×Q16, Dic10 [×2], C22×D5, C82Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, D5×D8, D5×Q16, C402Q8

Smallest permutation representation of C402Q8
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)
(1 186 265 311)(2 177 266 302)(3 168 267 293)(4 199 268 284)(5 190 269 315)(6 181 270 306)(7 172 271 297)(8 163 272 288)(9 194 273 319)(10 185 274 310)(11 176 275 301)(12 167 276 292)(13 198 277 283)(14 189 278 314)(15 180 279 305)(16 171 280 296)(17 162 241 287)(18 193 242 318)(19 184 243 309)(20 175 244 300)(21 166 245 291)(22 197 246 282)(23 188 247 313)(24 179 248 304)(25 170 249 295)(26 161 250 286)(27 192 251 317)(28 183 252 308)(29 174 253 299)(30 165 254 290)(31 196 255 281)(32 187 256 312)(33 178 257 303)(34 169 258 294)(35 200 259 285)(36 191 260 316)(37 182 261 307)(38 173 262 298)(39 164 263 289)(40 195 264 320)(41 235 143 102)(42 226 144 93)(43 217 145 84)(44 208 146 115)(45 239 147 106)(46 230 148 97)(47 221 149 88)(48 212 150 119)(49 203 151 110)(50 234 152 101)(51 225 153 92)(52 216 154 83)(53 207 155 114)(54 238 156 105)(55 229 157 96)(56 220 158 87)(57 211 159 118)(58 202 160 109)(59 233 121 100)(60 224 122 91)(61 215 123 82)(62 206 124 113)(63 237 125 104)(64 228 126 95)(65 219 127 86)(66 210 128 117)(67 201 129 108)(68 232 130 99)(69 223 131 90)(70 214 132 81)(71 205 133 112)(72 236 134 103)(73 227 135 94)(74 218 136 85)(75 209 137 116)(76 240 138 107)(77 231 139 98)(78 222 140 89)(79 213 141 120)(80 204 142 111)
(1 66 265 128)(2 75 266 137)(3 44 267 146)(4 53 268 155)(5 62 269 124)(6 71 270 133)(7 80 271 142)(8 49 272 151)(9 58 273 160)(10 67 274 129)(11 76 275 138)(12 45 276 147)(13 54 277 156)(14 63 278 125)(15 72 279 134)(16 41 280 143)(17 50 241 152)(18 59 242 121)(19 68 243 130)(20 77 244 139)(21 46 245 148)(22 55 246 157)(23 64 247 126)(24 73 248 135)(25 42 249 144)(26 51 250 153)(27 60 251 122)(28 69 252 131)(29 78 253 140)(30 47 254 149)(31 56 255 158)(32 65 256 127)(33 74 257 136)(34 43 258 145)(35 52 259 154)(36 61 260 123)(37 70 261 132)(38 79 262 141)(39 48 263 150)(40 57 264 159)(81 307 214 182)(82 316 215 191)(83 285 216 200)(84 294 217 169)(85 303 218 178)(86 312 219 187)(87 281 220 196)(88 290 221 165)(89 299 222 174)(90 308 223 183)(91 317 224 192)(92 286 225 161)(93 295 226 170)(94 304 227 179)(95 313 228 188)(96 282 229 197)(97 291 230 166)(98 300 231 175)(99 309 232 184)(100 318 233 193)(101 287 234 162)(102 296 235 171)(103 305 236 180)(104 314 237 189)(105 283 238 198)(106 292 239 167)(107 301 240 176)(108 310 201 185)(109 319 202 194)(110 288 203 163)(111 297 204 172)(112 306 205 181)(113 315 206 190)(114 284 207 199)(115 293 208 168)(116 302 209 177)(117 311 210 186)(118 320 211 195)(119 289 212 164)(120 298 213 173)

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

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

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

50 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444558888888810···102020202020···2040···40
size11112288101010104040222222101010102···244448···84···4

50 irreducible representations

dim1111112222222224444
type++++++--+++-++--++-
imageC1C2C2C2C2C2Q8Q8D4D5D8Q16D10D10Dic10Q8×D5D4×D5D5×D8D5×Q16
kernelC402Q8C10.D8C8×Dic5C405C4C5×C2.D8C20⋊Q8C52C8C40C2×Dic5C2.D8Dic5Dic5C4⋊C4C2×C8C8C4C22C2C2
# reps1211122222444282244

Matrix representation of C402Q8 in GL4(𝔽41) generated by

73500
7000
00177
00350
,
112800
223000
001238
002129
,
132600
252800
00404
00201
G:=sub<GL(4,GF(41))| [7,7,0,0,35,0,0,0,0,0,17,35,0,0,7,0],[11,22,0,0,28,30,0,0,0,0,12,21,0,0,38,29],[13,25,0,0,26,28,0,0,0,0,40,20,0,0,4,1] >;

C402Q8 in GAP, Magma, Sage, TeX

C_{40}\rtimes_2Q_8
% in TeX

G:=Group("C40:2Q8");
// GroupNames label

G:=SmallGroup(320,501);
// by ID

G=gap.SmallGroup(320,501);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,254,219,58,438,102,12550]);
// 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^9,c*b*c^-1=b^-1>;
// generators/relations

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