metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.6D4, C8.16D20, C10.4Q32, C10.4SD32, C20.5SD16, Dic20⋊11C4, C8.13(C4×D5), C40.51(C2×C4), (C2×C10).34D8, (C2×C20).92D4, C2.D8.3D5, C4.2(Q8⋊D5), C5⋊2(C2.Q32), (C2×C8).222D10, C2.2(D8.D5), C2.2(C5⋊Q32), (C2×C40).74C22, (C2×Dic20).8C2, C4.3(D10⋊C4), C2.8(D20⋊6C4), C20.50(C22⋊C4), C22.15(D4⋊D5), C10.21(D4⋊C4), (C5×C2.D8).3C2, (C2×C5⋊2C16).4C2, (C2×C4).116(C5⋊D4), SmallGroup(320,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.Q32
G = < a,b,c | a10=b16=1, c2=a5b8, bab-1=a-1, ac=ca, cbc-1=a5b-1 >
Subgroups: 238 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C16, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C2.D8, C2×C16, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C2.Q32, C5⋊2C16, Dic20, Dic20, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C5⋊2C16, C5×C2.D8, C2×Dic20, C10.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, SD32, Q32, C4×D5, D20, C5⋊D4, C2.Q32, D10⋊C4, D4⋊D5, Q8⋊D5, D20⋊6C4, D8.D5, C5⋊Q32, C10.Q32
►Regular action on 320 points
Generators in S320
(1 223 239 173 83 152 124 305 181 26)(2 27 182 306 125 153 84 174 240 224)(3 209 225 175 85 154 126 307 183 28)(4 29 184 308 127 155 86 176 226 210)(5 211 227 161 87 156 128 309 185 30)(6 31 186 310 113 157 88 162 228 212)(7 213 229 163 89 158 114 311 187 32)(8 17 188 312 115 159 90 164 230 214)(9 215 231 165 91 160 116 313 189 18)(10 19 190 314 117 145 92 166 232 216)(11 217 233 167 93 146 118 315 191 20)(12 21 192 316 119 147 94 168 234 218)(13 219 235 169 95 148 120 317 177 22)(14 23 178 318 121 149 96 170 236 220)(15 221 237 171 81 150 122 319 179 24)(16 25 180 320 123 151 82 172 238 222)(33 282 202 69 60 245 135 267 292 100)(34 101 293 268 136 246 61 70 203 283)(35 284 204 71 62 247 137 269 294 102)(36 103 295 270 138 248 63 72 205 285)(37 286 206 73 64 249 139 271 296 104)(38 105 297 272 140 250 49 74 207 287)(39 288 208 75 50 251 141 257 298 106)(40 107 299 258 142 252 51 76 193 273)(41 274 194 77 52 253 143 259 300 108)(42 109 301 260 144 254 53 78 195 275)(43 276 196 79 54 255 129 261 302 110)(44 111 303 262 130 256 55 80 197 277)(45 278 198 65 56 241 131 263 304 112)(46 97 289 264 132 242 57 66 199 279)(47 280 200 67 58 243 133 265 290 98)(48 99 291 266 134 244 59 68 201 281)
(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 46 160 250)(2 241 145 37)(3 44 146 248)(4 255 147 35)(5 42 148 246)(6 253 149 33)(7 40 150 244)(8 251 151 47)(9 38 152 242)(10 249 153 45)(11 36 154 256)(12 247 155 43)(13 34 156 254)(14 245 157 41)(15 48 158 252)(16 243 159 39)(17 141 82 280)(18 287 83 132)(19 139 84 278)(20 285 85 130)(21 137 86 276)(22 283 87 144)(23 135 88 274)(24 281 89 142)(25 133 90 288)(26 279 91 140)(27 131 92 286)(28 277 93 138)(29 129 94 284)(30 275 95 136)(31 143 96 282)(32 273 81 134)(49 223 97 116)(50 123 98 214)(51 221 99 114)(52 121 100 212)(53 219 101 128)(54 119 102 210)(55 217 103 126)(56 117 104 224)(57 215 105 124)(58 115 106 222)(59 213 107 122)(60 113 108 220)(61 211 109 120)(62 127 110 218)(63 209 111 118)(64 125 112 216)(65 314 296 240)(66 231 297 305)(67 312 298 238)(68 229 299 319)(69 310 300 236)(70 227 301 317)(71 308 302 234)(72 225 303 315)(73 306 304 232)(74 239 289 313)(75 320 290 230)(76 237 291 311)(77 318 292 228)(78 235 293 309)(79 316 294 226)(80 233 295 307)(161 260 177 203)(162 194 178 267)(163 258 179 201)(164 208 180 265)(165 272 181 199)(166 206 182 263)(167 270 183 197)(168 204 184 261)(169 268 185 195)(170 202 186 259)(171 266 187 193)(172 200 188 257)(173 264 189 207)(174 198 190 271)(175 262 191 205)(176 196 192 269)
G:=sub<Sym(320)| (1,223,239,173,83,152,124,305,181,26)(2,27,182,306,125,153,84,174,240,224)(3,209,225,175,85,154,126,307,183,28)(4,29,184,308,127,155,86,176,226,210)(5,211,227,161,87,156,128,309,185,30)(6,31,186,310,113,157,88,162,228,212)(7,213,229,163,89,158,114,311,187,32)(8,17,188,312,115,159,90,164,230,214)(9,215,231,165,91,160,116,313,189,18)(10,19,190,314,117,145,92,166,232,216)(11,217,233,167,93,146,118,315,191,20)(12,21,192,316,119,147,94,168,234,218)(13,219,235,169,95,148,120,317,177,22)(14,23,178,318,121,149,96,170,236,220)(15,221,237,171,81,150,122,319,179,24)(16,25,180,320,123,151,82,172,238,222)(33,282,202,69,60,245,135,267,292,100)(34,101,293,268,136,246,61,70,203,283)(35,284,204,71,62,247,137,269,294,102)(36,103,295,270,138,248,63,72,205,285)(37,286,206,73,64,249,139,271,296,104)(38,105,297,272,140,250,49,74,207,287)(39,288,208,75,50,251,141,257,298,106)(40,107,299,258,142,252,51,76,193,273)(41,274,194,77,52,253,143,259,300,108)(42,109,301,260,144,254,53,78,195,275)(43,276,196,79,54,255,129,261,302,110)(44,111,303,262,130,256,55,80,197,277)(45,278,198,65,56,241,131,263,304,112)(46,97,289,264,132,242,57,66,199,279)(47,280,200,67,58,243,133,265,290,98)(48,99,291,266,134,244,59,68,201,281), (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), (1,46,160,250)(2,241,145,37)(3,44,146,248)(4,255,147,35)(5,42,148,246)(6,253,149,33)(7,40,150,244)(8,251,151,47)(9,38,152,242)(10,249,153,45)(11,36,154,256)(12,247,155,43)(13,34,156,254)(14,245,157,41)(15,48,158,252)(16,243,159,39)(17,141,82,280)(18,287,83,132)(19,139,84,278)(20,285,85,130)(21,137,86,276)(22,283,87,144)(23,135,88,274)(24,281,89,142)(25,133,90,288)(26,279,91,140)(27,131,92,286)(28,277,93,138)(29,129,94,284)(30,275,95,136)(31,143,96,282)(32,273,81,134)(49,223,97,116)(50,123,98,214)(51,221,99,114)(52,121,100,212)(53,219,101,128)(54,119,102,210)(55,217,103,126)(56,117,104,224)(57,215,105,124)(58,115,106,222)(59,213,107,122)(60,113,108,220)(61,211,109,120)(62,127,110,218)(63,209,111,118)(64,125,112,216)(65,314,296,240)(66,231,297,305)(67,312,298,238)(68,229,299,319)(69,310,300,236)(70,227,301,317)(71,308,302,234)(72,225,303,315)(73,306,304,232)(74,239,289,313)(75,320,290,230)(76,237,291,311)(77,318,292,228)(78,235,293,309)(79,316,294,226)(80,233,295,307)(161,260,177,203)(162,194,178,267)(163,258,179,201)(164,208,180,265)(165,272,181,199)(166,206,182,263)(167,270,183,197)(168,204,184,261)(169,268,185,195)(170,202,186,259)(171,266,187,193)(172,200,188,257)(173,264,189,207)(174,198,190,271)(175,262,191,205)(176,196,192,269)>;
G:=Group( (1,223,239,173,83,152,124,305,181,26)(2,27,182,306,125,153,84,174,240,224)(3,209,225,175,85,154,126,307,183,28)(4,29,184,308,127,155,86,176,226,210)(5,211,227,161,87,156,128,309,185,30)(6,31,186,310,113,157,88,162,228,212)(7,213,229,163,89,158,114,311,187,32)(8,17,188,312,115,159,90,164,230,214)(9,215,231,165,91,160,116,313,189,18)(10,19,190,314,117,145,92,166,232,216)(11,217,233,167,93,146,118,315,191,20)(12,21,192,316,119,147,94,168,234,218)(13,219,235,169,95,148,120,317,177,22)(14,23,178,318,121,149,96,170,236,220)(15,221,237,171,81,150,122,319,179,24)(16,25,180,320,123,151,82,172,238,222)(33,282,202,69,60,245,135,267,292,100)(34,101,293,268,136,246,61,70,203,283)(35,284,204,71,62,247,137,269,294,102)(36,103,295,270,138,248,63,72,205,285)(37,286,206,73,64,249,139,271,296,104)(38,105,297,272,140,250,49,74,207,287)(39,288,208,75,50,251,141,257,298,106)(40,107,299,258,142,252,51,76,193,273)(41,274,194,77,52,253,143,259,300,108)(42,109,301,260,144,254,53,78,195,275)(43,276,196,79,54,255,129,261,302,110)(44,111,303,262,130,256,55,80,197,277)(45,278,198,65,56,241,131,263,304,112)(46,97,289,264,132,242,57,66,199,279)(47,280,200,67,58,243,133,265,290,98)(48,99,291,266,134,244,59,68,201,281), (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), (1,46,160,250)(2,241,145,37)(3,44,146,248)(4,255,147,35)(5,42,148,246)(6,253,149,33)(7,40,150,244)(8,251,151,47)(9,38,152,242)(10,249,153,45)(11,36,154,256)(12,247,155,43)(13,34,156,254)(14,245,157,41)(15,48,158,252)(16,243,159,39)(17,141,82,280)(18,287,83,132)(19,139,84,278)(20,285,85,130)(21,137,86,276)(22,283,87,144)(23,135,88,274)(24,281,89,142)(25,133,90,288)(26,279,91,140)(27,131,92,286)(28,277,93,138)(29,129,94,284)(30,275,95,136)(31,143,96,282)(32,273,81,134)(49,223,97,116)(50,123,98,214)(51,221,99,114)(52,121,100,212)(53,219,101,128)(54,119,102,210)(55,217,103,126)(56,117,104,224)(57,215,105,124)(58,115,106,222)(59,213,107,122)(60,113,108,220)(61,211,109,120)(62,127,110,218)(63,209,111,118)(64,125,112,216)(65,314,296,240)(66,231,297,305)(67,312,298,238)(68,229,299,319)(69,310,300,236)(70,227,301,317)(71,308,302,234)(72,225,303,315)(73,306,304,232)(74,239,289,313)(75,320,290,230)(76,237,291,311)(77,318,292,228)(78,235,293,309)(79,316,294,226)(80,233,295,307)(161,260,177,203)(162,194,178,267)(163,258,179,201)(164,208,180,265)(165,272,181,199)(166,206,182,263)(167,270,183,197)(168,204,184,261)(169,268,185,195)(170,202,186,259)(171,266,187,193)(172,200,188,257)(173,264,189,207)(174,198,190,271)(175,262,191,205)(176,196,192,269) );
G=PermutationGroup([[(1,223,239,173,83,152,124,305,181,26),(2,27,182,306,125,153,84,174,240,224),(3,209,225,175,85,154,126,307,183,28),(4,29,184,308,127,155,86,176,226,210),(5,211,227,161,87,156,128,309,185,30),(6,31,186,310,113,157,88,162,228,212),(7,213,229,163,89,158,114,311,187,32),(8,17,188,312,115,159,90,164,230,214),(9,215,231,165,91,160,116,313,189,18),(10,19,190,314,117,145,92,166,232,216),(11,217,233,167,93,146,118,315,191,20),(12,21,192,316,119,147,94,168,234,218),(13,219,235,169,95,148,120,317,177,22),(14,23,178,318,121,149,96,170,236,220),(15,221,237,171,81,150,122,319,179,24),(16,25,180,320,123,151,82,172,238,222),(33,282,202,69,60,245,135,267,292,100),(34,101,293,268,136,246,61,70,203,283),(35,284,204,71,62,247,137,269,294,102),(36,103,295,270,138,248,63,72,205,285),(37,286,206,73,64,249,139,271,296,104),(38,105,297,272,140,250,49,74,207,287),(39,288,208,75,50,251,141,257,298,106),(40,107,299,258,142,252,51,76,193,273),(41,274,194,77,52,253,143,259,300,108),(42,109,301,260,144,254,53,78,195,275),(43,276,196,79,54,255,129,261,302,110),(44,111,303,262,130,256,55,80,197,277),(45,278,198,65,56,241,131,263,304,112),(46,97,289,264,132,242,57,66,199,279),(47,280,200,67,58,243,133,265,290,98),(48,99,291,266,134,244,59,68,201,281)], [(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)], [(1,46,160,250),(2,241,145,37),(3,44,146,248),(4,255,147,35),(5,42,148,246),(6,253,149,33),(7,40,150,244),(8,251,151,47),(9,38,152,242),(10,249,153,45),(11,36,154,256),(12,247,155,43),(13,34,156,254),(14,245,157,41),(15,48,158,252),(16,243,159,39),(17,141,82,280),(18,287,83,132),(19,139,84,278),(20,285,85,130),(21,137,86,276),(22,283,87,144),(23,135,88,274),(24,281,89,142),(25,133,90,288),(26,279,91,140),(27,131,92,286),(28,277,93,138),(29,129,94,284),(30,275,95,136),(31,143,96,282),(32,273,81,134),(49,223,97,116),(50,123,98,214),(51,221,99,114),(52,121,100,212),(53,219,101,128),(54,119,102,210),(55,217,103,126),(56,117,104,224),(57,215,105,124),(58,115,106,222),(59,213,107,122),(60,113,108,220),(61,211,109,120),(62,127,110,218),(63,209,111,118),(64,125,112,216),(65,314,296,240),(66,231,297,305),(67,312,298,238),(68,229,299,319),(69,310,300,236),(70,227,301,317),(71,308,302,234),(72,225,303,315),(73,306,304,232),(74,239,289,313),(75,320,290,230),(76,237,291,311),(77,318,292,228),(78,235,293,309),(79,316,294,226),(80,233,295,307),(161,260,177,203),(162,194,178,267),(163,258,179,201),(164,208,180,265),(165,272,181,199),(166,206,182,263),(167,270,183,197),(168,204,184,261),(169,268,185,195),(170,202,186,259),(171,266,187,193),(172,200,188,257),(173,264,189,207),(174,198,190,271),(175,262,191,205),(176,196,192,269)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | SD16 | D8 | D10 | SD32 | Q32 | C4×D5 | D20 | C5⋊D4 | Q8⋊D5 | D4⋊D5 | D8.D5 | C5⋊Q32 |
| kernel | C10.Q32 | C2×C5⋊2C16 | C5×C2.D8 | C2×Dic20 | Dic20 | C40 | C2×C20 | C2.D8 | C20 | C2×C10 | C2×C8 | C10 | C10 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C10.Q32 ►in GL4(𝔽241) generated by
| 51 | 51 | 0 | 0 |
| 190 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 166 | 60 | 0 | 0 |
| 91 | 75 | 0 | 0 |
| 0 | 0 | 27 | 85 |
| 0 | 0 | 156 | 27 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 145 | 69 |
| 0 | 0 | 69 | 96 |
G:=sub<GL(4,GF(241))| [51,190,0,0,51,1,0,0,0,0,1,0,0,0,0,1],[166,91,0,0,60,75,0,0,0,0,27,156,0,0,85,27],[64,0,0,0,0,64,0,0,0,0,145,69,0,0,69,96] >;
C10.Q32 in GAP, Magma, Sage, TeX
C_{10}.Q_{32} % in TeX
G:=Group("C10.Q32"); // GroupNames label
G:=SmallGroup(320,50);
// by ID
G=gap.SmallGroup(320,50);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,675,346,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=1,c^2=a^5*b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations