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G = C405Q8order 320 = 26·5

5th semidirect product of C40 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C405Q8, C85Dic10, Dic5.10SD16, C52C85Q8, C52(C83Q8), C20⋊Q8.6C2, C4.20(Q8×D5), C4⋊C4.32D10, C4.Q8.5D5, C2.8(C20⋊Q8), C20.10(C2×Q8), (C2×C8).256D10, C10.13(C4⋊Q8), (C8×Dic5).6C2, C406C4.13C2, C2.21(D5×SD16), C4.20(C2×Dic10), C10.36(C2×SD16), C22.210(D4×D5), C20.Q8.5C2, (C2×C40).157C22, (C2×C20).271C23, (C2×Dic5).141D4, C4⋊Dic5.103C22, (C4×Dic5).260C22, (C5×C4.Q8).4C2, (C2×C10).276(C2×D4), (C5×C4⋊C4).64C22, (C2×C4).374(C22×D5), (C2×C52C8).232C22, SmallGroup(320,482)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C405Q8
C1C5C10C2×C10C2×C20C4×Dic5C8×Dic5 — C405Q8
C5C10C2×C20 — C405Q8
C1C22C2×C4C4.Q8

Generators and relations for C405Q8
 G = < a,b,c | a40=b4=1, c2=b2, bab-1=a11, cac-1=a9, cbc-1=b-1 >

Subgroups: 382 in 98 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], 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, C4.Q8, C4.Q8 [×3], C4⋊Q8 [×2], C52C8 [×2], C40 [×2], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C83Q8, C2×C52C8, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10 [×2], C20.Q8 [×2], C8×Dic5, C406C4, C5×C4.Q8, C20⋊Q8 [×2], C405Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, SD16 [×4], C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C2×SD16 [×2], Dic10 [×2], C22×D5, C83Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, D5×SD16 [×2], C405Q8

Smallest permutation representation of C405Q8
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)
(1 195 294 279)(2 166 295 250)(3 177 296 261)(4 188 297 272)(5 199 298 243)(6 170 299 254)(7 181 300 265)(8 192 301 276)(9 163 302 247)(10 174 303 258)(11 185 304 269)(12 196 305 280)(13 167 306 251)(14 178 307 262)(15 189 308 273)(16 200 309 244)(17 171 310 255)(18 182 311 266)(19 193 312 277)(20 164 313 248)(21 175 314 259)(22 186 315 270)(23 197 316 241)(24 168 317 252)(25 179 318 263)(26 190 319 274)(27 161 320 245)(28 172 281 256)(29 183 282 267)(30 194 283 278)(31 165 284 249)(32 176 285 260)(33 187 286 271)(34 198 287 242)(35 169 288 253)(36 180 289 264)(37 191 290 275)(38 162 291 246)(39 173 292 257)(40 184 293 268)(41 226 120 126)(42 237 81 137)(43 208 82 148)(44 219 83 159)(45 230 84 130)(46 201 85 141)(47 212 86 152)(48 223 87 123)(49 234 88 134)(50 205 89 145)(51 216 90 156)(52 227 91 127)(53 238 92 138)(54 209 93 149)(55 220 94 160)(56 231 95 131)(57 202 96 142)(58 213 97 153)(59 224 98 124)(60 235 99 135)(61 206 100 146)(62 217 101 157)(63 228 102 128)(64 239 103 139)(65 210 104 150)(66 221 105 121)(67 232 106 132)(68 203 107 143)(69 214 108 154)(70 225 109 125)(71 236 110 136)(72 207 111 147)(73 218 112 158)(74 229 113 129)(75 240 114 140)(76 211 115 151)(77 222 116 122)(78 233 117 133)(79 204 118 144)(80 215 119 155)
(1 240 294 140)(2 209 295 149)(3 218 296 158)(4 227 297 127)(5 236 298 136)(6 205 299 145)(7 214 300 154)(8 223 301 123)(9 232 302 132)(10 201 303 141)(11 210 304 150)(12 219 305 159)(13 228 306 128)(14 237 307 137)(15 206 308 146)(16 215 309 155)(17 224 310 124)(18 233 311 133)(19 202 312 142)(20 211 313 151)(21 220 314 160)(22 229 315 129)(23 238 316 138)(24 207 317 147)(25 216 318 156)(26 225 319 125)(27 234 320 134)(28 203 281 143)(29 212 282 152)(30 221 283 121)(31 230 284 130)(32 239 285 139)(33 208 286 148)(34 217 287 157)(35 226 288 126)(36 235 289 135)(37 204 290 144)(38 213 291 153)(39 222 292 122)(40 231 293 131)(41 253 120 169)(42 262 81 178)(43 271 82 187)(44 280 83 196)(45 249 84 165)(46 258 85 174)(47 267 86 183)(48 276 87 192)(49 245 88 161)(50 254 89 170)(51 263 90 179)(52 272 91 188)(53 241 92 197)(54 250 93 166)(55 259 94 175)(56 268 95 184)(57 277 96 193)(58 246 97 162)(59 255 98 171)(60 264 99 180)(61 273 100 189)(62 242 101 198)(63 251 102 167)(64 260 103 176)(65 269 104 185)(66 278 105 194)(67 247 106 163)(68 256 107 172)(69 265 108 181)(70 274 109 190)(71 243 110 199)(72 252 111 168)(73 261 112 177)(74 270 113 186)(75 279 114 195)(76 248 115 164)(77 257 116 173)(78 266 117 182)(79 275 118 191)(80 244 119 200)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim11111122222222444
type++++++--++++--+
imageC1C2C2C2C2C2Q8Q8D4D5SD16D10D10Dic10Q8×D5D4×D5D5×SD16
kernelC405Q8C20.Q8C8×Dic5C406C4C5×C4.Q8C20⋊Q8C52C8C40C2×Dic5C4.Q8Dic5C4⋊C4C2×C8C8C4C22C2
# reps12111222228428228

Matrix representation of C405Q8 in GL4(𝔽41) generated by

261500
262600
00034
0066
,
21300
32000
00232
003739
,
04000
1000
00203
00321
G:=sub<GL(4,GF(41))| [26,26,0,0,15,26,0,0,0,0,0,6,0,0,34,6],[21,3,0,0,3,20,0,0,0,0,2,37,0,0,32,39],[0,1,0,0,40,0,0,0,0,0,20,3,0,0,3,21] >;

C405Q8 in GAP, Magma, Sage, TeX

C_{40}\rtimes_5Q_8
% in TeX

G:=Group("C40:5Q8");
// GroupNames label

G:=SmallGroup(320,482);
// by ID

G=gap.SmallGroup(320,482);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,254,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations

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