direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C10×Q32, C40.74D4, C20.46D8, C40.74C23, C80.22C22, C4.8(C5×D8), C4.9(D4×C10), C8.11(C5×D4), (C2×C16).4C10, (C2×C80).10C2, C16.5(C2×C10), (C2×C10).57D8, C2.14(C10×D8), C10.86(C2×D8), (C2×C20).428D4, C20.316(C2×D4), C8.5(C22×C10), Q16.1(C2×C10), (C2×Q16).4C10, C22.16(C5×D8), (C10×Q16).11C2, (C2×C40).428C22, (C5×Q16).13C22, (C2×C4).84(C5×D4), (C2×C8).86(C2×C10), SmallGroup(320,1008)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 146 in 82 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], C16 [×2], C2×C8, Q16 [×4], Q16 [×2], C2×Q8 [×2], C20 [×2], C20 [×4], C2×C10, C2×C16, Q32 [×4], C2×Q16 [×2], C40 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×6], C2×Q32, C80 [×2], C2×C40, C5×Q16 [×4], C5×Q16 [×2], Q8×C10 [×2], C2×C80, C5×Q32 [×4], C10×Q16 [×2], C10×Q32
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], Q32 [×2], C2×D8, C5×D4 [×2], C22×C10, C2×Q32, C5×D8 [×2], D4×C10, C5×Q32 [×2], C10×D8, C10×Q32
Generators and relations
G = < a,b,c | a10=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
►Regular action on 320 points
Generators in S320
(1 220 72 301 91 113 144 280 52 166)(2 221 73 302 92 114 129 281 53 167)(3 222 74 303 93 115 130 282 54 168)(4 223 75 304 94 116 131 283 55 169)(5 224 76 289 95 117 132 284 56 170)(6 209 77 290 96 118 133 285 57 171)(7 210 78 291 81 119 134 286 58 172)(8 211 79 292 82 120 135 287 59 173)(9 212 80 293 83 121 136 288 60 174)(10 213 65 294 84 122 137 273 61 175)(11 214 66 295 85 123 138 274 62 176)(12 215 67 296 86 124 139 275 63 161)(13 216 68 297 87 125 140 276 64 162)(14 217 69 298 88 126 141 277 49 163)(15 218 70 299 89 127 142 278 50 164)(16 219 71 300 90 128 143 279 51 165)(17 181 317 47 225 145 243 271 105 198)(18 182 318 48 226 146 244 272 106 199)(19 183 319 33 227 147 245 257 107 200)(20 184 320 34 228 148 246 258 108 201)(21 185 305 35 229 149 247 259 109 202)(22 186 306 36 230 150 248 260 110 203)(23 187 307 37 231 151 249 261 111 204)(24 188 308 38 232 152 250 262 112 205)(25 189 309 39 233 153 251 263 97 206)(26 190 310 40 234 154 252 264 98 207)(27 191 311 41 235 155 253 265 99 208)(28 192 312 42 236 156 254 266 100 193)(29 177 313 43 237 157 255 267 101 194)(30 178 314 44 238 158 256 268 102 195)(31 179 315 45 239 159 241 269 103 196)(32 180 316 46 240 160 242 270 104 197)
(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)
(1 314 9 306)(2 313 10 305)(3 312 11 320)(4 311 12 319)(5 310 13 318)(6 309 14 317)(7 308 15 316)(8 307 16 315)(17 57 25 49)(18 56 26 64)(19 55 27 63)(20 54 28 62)(21 53 29 61)(22 52 30 60)(23 51 31 59)(24 50 32 58)(33 223 41 215)(34 222 42 214)(35 221 43 213)(36 220 44 212)(37 219 45 211)(38 218 46 210)(39 217 47 209)(40 216 48 224)(65 229 73 237)(66 228 74 236)(67 227 75 235)(68 226 76 234)(69 225 77 233)(70 240 78 232)(71 239 79 231)(72 238 80 230)(81 250 89 242)(82 249 90 241)(83 248 91 256)(84 247 92 255)(85 246 93 254)(86 245 94 253)(87 244 95 252)(88 243 96 251)(97 141 105 133)(98 140 106 132)(99 139 107 131)(100 138 108 130)(101 137 109 129)(102 136 110 144)(103 135 111 143)(104 134 112 142)(113 268 121 260)(114 267 122 259)(115 266 123 258)(116 265 124 257)(117 264 125 272)(118 263 126 271)(119 262 127 270)(120 261 128 269)(145 290 153 298)(146 289 154 297)(147 304 155 296)(148 303 156 295)(149 302 157 294)(150 301 158 293)(151 300 159 292)(152 299 160 291)(161 183 169 191)(162 182 170 190)(163 181 171 189)(164 180 172 188)(165 179 173 187)(166 178 174 186)(167 177 175 185)(168 192 176 184)(193 274 201 282)(194 273 202 281)(195 288 203 280)(196 287 204 279)(197 286 205 278)(198 285 206 277)(199 284 207 276)(200 283 208 275)
G:=sub<Sym(320)| (1,220,72,301,91,113,144,280,52,166)(2,221,73,302,92,114,129,281,53,167)(3,222,74,303,93,115,130,282,54,168)(4,223,75,304,94,116,131,283,55,169)(5,224,76,289,95,117,132,284,56,170)(6,209,77,290,96,118,133,285,57,171)(7,210,78,291,81,119,134,286,58,172)(8,211,79,292,82,120,135,287,59,173)(9,212,80,293,83,121,136,288,60,174)(10,213,65,294,84,122,137,273,61,175)(11,214,66,295,85,123,138,274,62,176)(12,215,67,296,86,124,139,275,63,161)(13,216,68,297,87,125,140,276,64,162)(14,217,69,298,88,126,141,277,49,163)(15,218,70,299,89,127,142,278,50,164)(16,219,71,300,90,128,143,279,51,165)(17,181,317,47,225,145,243,271,105,198)(18,182,318,48,226,146,244,272,106,199)(19,183,319,33,227,147,245,257,107,200)(20,184,320,34,228,148,246,258,108,201)(21,185,305,35,229,149,247,259,109,202)(22,186,306,36,230,150,248,260,110,203)(23,187,307,37,231,151,249,261,111,204)(24,188,308,38,232,152,250,262,112,205)(25,189,309,39,233,153,251,263,97,206)(26,190,310,40,234,154,252,264,98,207)(27,191,311,41,235,155,253,265,99,208)(28,192,312,42,236,156,254,266,100,193)(29,177,313,43,237,157,255,267,101,194)(30,178,314,44,238,158,256,268,102,195)(31,179,315,45,239,159,241,269,103,196)(32,180,316,46,240,160,242,270,104,197), (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), (1,314,9,306)(2,313,10,305)(3,312,11,320)(4,311,12,319)(5,310,13,318)(6,309,14,317)(7,308,15,316)(8,307,16,315)(17,57,25,49)(18,56,26,64)(19,55,27,63)(20,54,28,62)(21,53,29,61)(22,52,30,60)(23,51,31,59)(24,50,32,58)(33,223,41,215)(34,222,42,214)(35,221,43,213)(36,220,44,212)(37,219,45,211)(38,218,46,210)(39,217,47,209)(40,216,48,224)(65,229,73,237)(66,228,74,236)(67,227,75,235)(68,226,76,234)(69,225,77,233)(70,240,78,232)(71,239,79,231)(72,238,80,230)(81,250,89,242)(82,249,90,241)(83,248,91,256)(84,247,92,255)(85,246,93,254)(86,245,94,253)(87,244,95,252)(88,243,96,251)(97,141,105,133)(98,140,106,132)(99,139,107,131)(100,138,108,130)(101,137,109,129)(102,136,110,144)(103,135,111,143)(104,134,112,142)(113,268,121,260)(114,267,122,259)(115,266,123,258)(116,265,124,257)(117,264,125,272)(118,263,126,271)(119,262,127,270)(120,261,128,269)(145,290,153,298)(146,289,154,297)(147,304,155,296)(148,303,156,295)(149,302,157,294)(150,301,158,293)(151,300,159,292)(152,299,160,291)(161,183,169,191)(162,182,170,190)(163,181,171,189)(164,180,172,188)(165,179,173,187)(166,178,174,186)(167,177,175,185)(168,192,176,184)(193,274,201,282)(194,273,202,281)(195,288,203,280)(196,287,204,279)(197,286,205,278)(198,285,206,277)(199,284,207,276)(200,283,208,275)>;
G:=Group( (1,220,72,301,91,113,144,280,52,166)(2,221,73,302,92,114,129,281,53,167)(3,222,74,303,93,115,130,282,54,168)(4,223,75,304,94,116,131,283,55,169)(5,224,76,289,95,117,132,284,56,170)(6,209,77,290,96,118,133,285,57,171)(7,210,78,291,81,119,134,286,58,172)(8,211,79,292,82,120,135,287,59,173)(9,212,80,293,83,121,136,288,60,174)(10,213,65,294,84,122,137,273,61,175)(11,214,66,295,85,123,138,274,62,176)(12,215,67,296,86,124,139,275,63,161)(13,216,68,297,87,125,140,276,64,162)(14,217,69,298,88,126,141,277,49,163)(15,218,70,299,89,127,142,278,50,164)(16,219,71,300,90,128,143,279,51,165)(17,181,317,47,225,145,243,271,105,198)(18,182,318,48,226,146,244,272,106,199)(19,183,319,33,227,147,245,257,107,200)(20,184,320,34,228,148,246,258,108,201)(21,185,305,35,229,149,247,259,109,202)(22,186,306,36,230,150,248,260,110,203)(23,187,307,37,231,151,249,261,111,204)(24,188,308,38,232,152,250,262,112,205)(25,189,309,39,233,153,251,263,97,206)(26,190,310,40,234,154,252,264,98,207)(27,191,311,41,235,155,253,265,99,208)(28,192,312,42,236,156,254,266,100,193)(29,177,313,43,237,157,255,267,101,194)(30,178,314,44,238,158,256,268,102,195)(31,179,315,45,239,159,241,269,103,196)(32,180,316,46,240,160,242,270,104,197), (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), (1,314,9,306)(2,313,10,305)(3,312,11,320)(4,311,12,319)(5,310,13,318)(6,309,14,317)(7,308,15,316)(8,307,16,315)(17,57,25,49)(18,56,26,64)(19,55,27,63)(20,54,28,62)(21,53,29,61)(22,52,30,60)(23,51,31,59)(24,50,32,58)(33,223,41,215)(34,222,42,214)(35,221,43,213)(36,220,44,212)(37,219,45,211)(38,218,46,210)(39,217,47,209)(40,216,48,224)(65,229,73,237)(66,228,74,236)(67,227,75,235)(68,226,76,234)(69,225,77,233)(70,240,78,232)(71,239,79,231)(72,238,80,230)(81,250,89,242)(82,249,90,241)(83,248,91,256)(84,247,92,255)(85,246,93,254)(86,245,94,253)(87,244,95,252)(88,243,96,251)(97,141,105,133)(98,140,106,132)(99,139,107,131)(100,138,108,130)(101,137,109,129)(102,136,110,144)(103,135,111,143)(104,134,112,142)(113,268,121,260)(114,267,122,259)(115,266,123,258)(116,265,124,257)(117,264,125,272)(118,263,126,271)(119,262,127,270)(120,261,128,269)(145,290,153,298)(146,289,154,297)(147,304,155,296)(148,303,156,295)(149,302,157,294)(150,301,158,293)(151,300,159,292)(152,299,160,291)(161,183,169,191)(162,182,170,190)(163,181,171,189)(164,180,172,188)(165,179,173,187)(166,178,174,186)(167,177,175,185)(168,192,176,184)(193,274,201,282)(194,273,202,281)(195,288,203,280)(196,287,204,279)(197,286,205,278)(198,285,206,277)(199,284,207,276)(200,283,208,275) );
G=PermutationGroup([(1,220,72,301,91,113,144,280,52,166),(2,221,73,302,92,114,129,281,53,167),(3,222,74,303,93,115,130,282,54,168),(4,223,75,304,94,116,131,283,55,169),(5,224,76,289,95,117,132,284,56,170),(6,209,77,290,96,118,133,285,57,171),(7,210,78,291,81,119,134,286,58,172),(8,211,79,292,82,120,135,287,59,173),(9,212,80,293,83,121,136,288,60,174),(10,213,65,294,84,122,137,273,61,175),(11,214,66,295,85,123,138,274,62,176),(12,215,67,296,86,124,139,275,63,161),(13,216,68,297,87,125,140,276,64,162),(14,217,69,298,88,126,141,277,49,163),(15,218,70,299,89,127,142,278,50,164),(16,219,71,300,90,128,143,279,51,165),(17,181,317,47,225,145,243,271,105,198),(18,182,318,48,226,146,244,272,106,199),(19,183,319,33,227,147,245,257,107,200),(20,184,320,34,228,148,246,258,108,201),(21,185,305,35,229,149,247,259,109,202),(22,186,306,36,230,150,248,260,110,203),(23,187,307,37,231,151,249,261,111,204),(24,188,308,38,232,152,250,262,112,205),(25,189,309,39,233,153,251,263,97,206),(26,190,310,40,234,154,252,264,98,207),(27,191,311,41,235,155,253,265,99,208),(28,192,312,42,236,156,254,266,100,193),(29,177,313,43,237,157,255,267,101,194),(30,178,314,44,238,158,256,268,102,195),(31,179,315,45,239,159,241,269,103,196),(32,180,316,46,240,160,242,270,104,197)], [(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)], [(1,314,9,306),(2,313,10,305),(3,312,11,320),(4,311,12,319),(5,310,13,318),(6,309,14,317),(7,308,15,316),(8,307,16,315),(17,57,25,49),(18,56,26,64),(19,55,27,63),(20,54,28,62),(21,53,29,61),(22,52,30,60),(23,51,31,59),(24,50,32,58),(33,223,41,215),(34,222,42,214),(35,221,43,213),(36,220,44,212),(37,219,45,211),(38,218,46,210),(39,217,47,209),(40,216,48,224),(65,229,73,237),(66,228,74,236),(67,227,75,235),(68,226,76,234),(69,225,77,233),(70,240,78,232),(71,239,79,231),(72,238,80,230),(81,250,89,242),(82,249,90,241),(83,248,91,256),(84,247,92,255),(85,246,93,254),(86,245,94,253),(87,244,95,252),(88,243,96,251),(97,141,105,133),(98,140,106,132),(99,139,107,131),(100,138,108,130),(101,137,109,129),(102,136,110,144),(103,135,111,143),(104,134,112,142),(113,268,121,260),(114,267,122,259),(115,266,123,258),(116,265,124,257),(117,264,125,272),(118,263,126,271),(119,262,127,270),(120,261,128,269),(145,290,153,298),(146,289,154,297),(147,304,155,296),(148,303,156,295),(149,302,157,294),(150,301,158,293),(151,300,159,292),(152,299,160,291),(161,183,169,191),(162,182,170,190),(163,181,171,189),(164,180,172,188),(165,179,173,187),(166,178,174,186),(167,177,175,185),(168,192,176,184),(193,274,201,282),(194,273,202,281),(195,288,203,280),(196,287,204,279),(197,286,205,278),(198,285,206,277),(199,284,207,276),(200,283,208,275)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 1 | 0 | 0 |
| 0 | 85 | 214 |
| 0 | 27 | 85 |
| 1 | 0 | 0 |
| 0 | 31 | 22 |
| 0 | 22 | 210 |
G:=sub<GL(3,GF(241))| [240,0,0,0,87,0,0,0,87],[1,0,0,0,85,27,0,214,85],[1,0,0,0,31,22,0,22,210] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20X | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | Q32 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | C5×Q32 |
| kernel | C10×Q32 | C2×C80 | C5×Q32 | C10×Q16 | C2×Q32 | C2×C16 | Q32 | C2×Q16 | C40 | C2×C20 | C20 | C2×C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times Q_{32} % in TeX
G:=Group("C10xQ32"); // GroupNames label
G:=SmallGroup(320,1008);
// by ID
G=gap.SmallGroup(320,1008);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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