direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C42.30C22, C4⋊Q8.8C10, C8⋊C4.6C10, (C2×C20).342D4, C42.28(C2×C10), Q8⋊C4.7C10, C42.C2.3C10, C20.273(C4○D4), (C2×C40).337C22, (C4×C20).270C22, (C2×C20).947C23, C22.112(D4×C10), C10.76(C4.4D4), (Q8×C10).175C22, C10.146(C8.C22), C4.18(C5×C4○D4), (C2×C4).43(C5×D4), (C5×C4⋊Q8).23C2, C4⋊C4.22(C2×C10), (C2×C8).58(C2×C10), (C5×C8⋊C4).12C2, (C2×C10).668(C2×D4), (C2×Q8).19(C2×C10), C2.14(C5×C4.4D4), C2.21(C5×C8.C22), (C5×C4⋊C4).242C22, (C5×Q8⋊C4).16C2, (C5×C42.C2).10C2, (C2×C4).122(C22×C10), SmallGroup(320,992)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42.30C22
G = < a,b,c,d,e | a5=b4=c4=1, d2=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, ebe-1=bc2, dcd-1=c-1, ce=ec, ede-1=b2c-1d >
Subgroups: 146 in 90 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C42.30C22, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C5×C8⋊C4, C5×Q8⋊C4, C5×C42.C2, C5×C4⋊Q8, C5×C42.30C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4.4D4, C8.C22, C5×D4, C22×C10, C42.30C22, D4×C10, C5×C4○D4, C5×C4.4D4, C5×C8.C22, C5×C42.30C22
►Regular action on 320 points
Generators in S320
(1 107 27 99 19)(2 108 28 100 20)(3 109 29 101 21)(4 110 30 102 22)(5 111 31 103 23)(6 112 32 104 24)(7 105 25 97 17)(8 106 26 98 18)(9 46 118 38 90)(10 47 119 39 91)(11 48 120 40 92)(12 41 113 33 93)(13 42 114 34 94)(14 43 115 35 95)(15 44 116 36 96)(16 45 117 37 89)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 165 153 73 145)(66 166 154 74 146)(67 167 155 75 147)(68 168 156 76 148)(69 161 157 77 149)(70 162 158 78 150)(71 163 159 79 151)(72 164 160 80 152)(81 318 234 310 226)(82 319 235 311 227)(83 320 236 312 228)(84 313 237 305 229)(85 314 238 306 230)(86 315 239 307 231)(87 316 240 308 232)(88 317 233 309 225)(169 245 261 177 253)(170 246 262 178 254)(171 247 263 179 255)(172 248 264 180 256)(173 241 257 181 249)(174 242 258 182 250)(175 243 259 183 251)(176 244 260 184 252)(185 201 277 193 269)(186 202 278 194 270)(187 203 279 195 271)(188 204 280 196 272)(189 205 273 197 265)(190 206 274 198 266)(191 207 275 199 267)(192 208 276 200 268)(209 285 301 217 293)(210 286 302 218 294)(211 287 303 219 295)(212 288 304 220 296)(213 281 297 221 289)(214 282 298 222 290)(215 283 299 223 291)(216 284 300 224 292)
(1 68 125 34)(2 65 126 39)(3 70 127 36)(4 67 128 33)(5 72 121 38)(6 69 122 35)(7 66 123 40)(8 71 124 37)(9 31 160 57)(10 28 153 62)(11 25 154 59)(12 30 155 64)(13 27 156 61)(14 32 157 58)(15 29 158 63)(16 26 159 60)(17 146 51 120)(18 151 52 117)(19 148 53 114)(20 145 54 119)(21 150 55 116)(22 147 56 113)(23 152 49 118)(24 149 50 115)(41 102 75 136)(42 99 76 133)(43 104 77 130)(44 101 78 135)(45 98 79 132)(46 103 80 129)(47 100 73 134)(48 97 74 131)(81 297 207 262)(82 302 208 259)(83 299 201 264)(84 304 202 261)(85 301 203 258)(86 298 204 263)(87 303 205 260)(88 300 206 257)(89 106 163 140)(90 111 164 137)(91 108 165 142)(92 105 166 139)(93 110 167 144)(94 107 168 141)(95 112 161 138)(96 109 162 143)(169 305 212 270)(170 310 213 267)(171 307 214 272)(172 312 215 269)(173 309 216 266)(174 306 209 271)(175 311 210 268)(176 308 211 265)(177 313 220 278)(178 318 221 275)(179 315 222 280)(180 320 223 277)(181 317 224 274)(182 314 217 279)(183 319 218 276)(184 316 219 273)(185 248 228 283)(186 245 229 288)(187 242 230 285)(188 247 231 282)(189 244 232 287)(190 241 225 284)(191 246 226 281)(192 243 227 286)(193 256 236 291)(194 253 237 296)(195 250 238 293)(196 255 239 290)(197 252 240 295)(198 249 233 292)(199 254 234 289)(200 251 235 294)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 195 197 199)(194 196 198 200)(201 203 205 207)(202 204 206 208)(209 211 213 215)(210 212 214 216)(217 219 221 223)(218 220 222 224)(225 227 229 231)(226 228 230 232)(233 235 237 239)(234 236 238 240)(241 243 245 247)(242 244 246 248)(249 251 253 255)(250 252 254 256)(257 259 261 263)(258 260 262 264)(265 267 269 271)(266 268 270 272)(273 275 277 279)(274 276 278 280)(281 283 285 287)(282 284 286 288)(289 291 293 295)(290 292 294 296)(297 299 301 303)(298 300 302 304)(305 307 309 311)(306 308 310 312)(313 315 317 319)(314 316 318 320)
(1 283 5 287)(2 243 6 247)(3 281 7 285)(4 241 8 245)(9 273 13 277)(10 319 14 315)(11 279 15 275)(12 317 16 313)(17 209 21 213)(18 169 22 173)(19 215 23 211)(20 175 24 171)(25 217 29 221)(26 177 30 181)(27 223 31 219)(28 183 32 179)(33 225 37 229)(34 185 38 189)(35 231 39 227)(36 191 40 187)(41 233 45 237)(42 193 46 197)(43 239 47 235)(44 199 48 195)(49 176 53 172)(50 214 54 210)(51 174 55 170)(52 212 56 216)(57 184 61 180)(58 222 62 218)(59 182 63 178)(60 220 64 224)(65 192 69 188)(66 230 70 226)(67 190 71 186)(68 228 72 232)(73 200 77 196)(74 238 78 234)(75 198 79 194)(76 236 80 240)(81 166 85 162)(82 95 86 91)(83 164 87 168)(84 93 88 89)(90 205 94 201)(92 203 96 207)(97 293 101 289)(98 253 102 249)(99 291 103 295)(100 251 104 255)(105 301 109 297)(106 261 110 257)(107 299 111 303)(108 259 112 263)(113 309 117 305)(114 269 118 265)(115 307 119 311)(116 267 120 271)(121 244 125 248)(122 282 126 286)(123 242 127 246)(124 288 128 284)(129 252 133 256)(130 290 134 294)(131 250 135 254)(132 296 136 292)(137 260 141 264)(138 298 142 302)(139 258 143 262)(140 304 144 300)(145 268 149 272)(146 306 150 310)(147 266 151 270)(148 312 152 308)(153 276 157 280)(154 314 158 318)(155 274 159 278)(156 320 160 316)(161 204 165 208)(163 202 167 206)
(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)
G:=sub<Sym(320)| (1,107,27,99,19)(2,108,28,100,20)(3,109,29,101,21)(4,110,30,102,22)(5,111,31,103,23)(6,112,32,104,24)(7,105,25,97,17)(8,106,26,98,18)(9,46,118,38,90)(10,47,119,39,91)(11,48,120,40,92)(12,41,113,33,93)(13,42,114,34,94)(14,43,115,35,95)(15,44,116,36,96)(16,45,117,37,89)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,165,153,73,145)(66,166,154,74,146)(67,167,155,75,147)(68,168,156,76,148)(69,161,157,77,149)(70,162,158,78,150)(71,163,159,79,151)(72,164,160,80,152)(81,318,234,310,226)(82,319,235,311,227)(83,320,236,312,228)(84,313,237,305,229)(85,314,238,306,230)(86,315,239,307,231)(87,316,240,308,232)(88,317,233,309,225)(169,245,261,177,253)(170,246,262,178,254)(171,247,263,179,255)(172,248,264,180,256)(173,241,257,181,249)(174,242,258,182,250)(175,243,259,183,251)(176,244,260,184,252)(185,201,277,193,269)(186,202,278,194,270)(187,203,279,195,271)(188,204,280,196,272)(189,205,273,197,265)(190,206,274,198,266)(191,207,275,199,267)(192,208,276,200,268)(209,285,301,217,293)(210,286,302,218,294)(211,287,303,219,295)(212,288,304,220,296)(213,281,297,221,289)(214,282,298,222,290)(215,283,299,223,291)(216,284,300,224,292), (1,68,125,34)(2,65,126,39)(3,70,127,36)(4,67,128,33)(5,72,121,38)(6,69,122,35)(7,66,123,40)(8,71,124,37)(9,31,160,57)(10,28,153,62)(11,25,154,59)(12,30,155,64)(13,27,156,61)(14,32,157,58)(15,29,158,63)(16,26,159,60)(17,146,51,120)(18,151,52,117)(19,148,53,114)(20,145,54,119)(21,150,55,116)(22,147,56,113)(23,152,49,118)(24,149,50,115)(41,102,75,136)(42,99,76,133)(43,104,77,130)(44,101,78,135)(45,98,79,132)(46,103,80,129)(47,100,73,134)(48,97,74,131)(81,297,207,262)(82,302,208,259)(83,299,201,264)(84,304,202,261)(85,301,203,258)(86,298,204,263)(87,303,205,260)(88,300,206,257)(89,106,163,140)(90,111,164,137)(91,108,165,142)(92,105,166,139)(93,110,167,144)(94,107,168,141)(95,112,161,138)(96,109,162,143)(169,305,212,270)(170,310,213,267)(171,307,214,272)(172,312,215,269)(173,309,216,266)(174,306,209,271)(175,311,210,268)(176,308,211,265)(177,313,220,278)(178,318,221,275)(179,315,222,280)(180,320,223,277)(181,317,224,274)(182,314,217,279)(183,319,218,276)(184,316,219,273)(185,248,228,283)(186,245,229,288)(187,242,230,285)(188,247,231,282)(189,244,232,287)(190,241,225,284)(191,246,226,281)(192,243,227,286)(193,256,236,291)(194,253,237,296)(195,250,238,293)(196,255,239,290)(197,252,240,295)(198,249,233,292)(199,254,234,289)(200,251,235,294), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,283,5,287)(2,243,6,247)(3,281,7,285)(4,241,8,245)(9,273,13,277)(10,319,14,315)(11,279,15,275)(12,317,16,313)(17,209,21,213)(18,169,22,173)(19,215,23,211)(20,175,24,171)(25,217,29,221)(26,177,30,181)(27,223,31,219)(28,183,32,179)(33,225,37,229)(34,185,38,189)(35,231,39,227)(36,191,40,187)(41,233,45,237)(42,193,46,197)(43,239,47,235)(44,199,48,195)(49,176,53,172)(50,214,54,210)(51,174,55,170)(52,212,56,216)(57,184,61,180)(58,222,62,218)(59,182,63,178)(60,220,64,224)(65,192,69,188)(66,230,70,226)(67,190,71,186)(68,228,72,232)(73,200,77,196)(74,238,78,234)(75,198,79,194)(76,236,80,240)(81,166,85,162)(82,95,86,91)(83,164,87,168)(84,93,88,89)(90,205,94,201)(92,203,96,207)(97,293,101,289)(98,253,102,249)(99,291,103,295)(100,251,104,255)(105,301,109,297)(106,261,110,257)(107,299,111,303)(108,259,112,263)(113,309,117,305)(114,269,118,265)(115,307,119,311)(116,267,120,271)(121,244,125,248)(122,282,126,286)(123,242,127,246)(124,288,128,284)(129,252,133,256)(130,290,134,294)(131,250,135,254)(132,296,136,292)(137,260,141,264)(138,298,142,302)(139,258,143,262)(140,304,144,300)(145,268,149,272)(146,306,150,310)(147,266,151,270)(148,312,152,308)(153,276,157,280)(154,314,158,318)(155,274,159,278)(156,320,160,316)(161,204,165,208)(163,202,167,206), (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)>;
G:=Group( (1,107,27,99,19)(2,108,28,100,20)(3,109,29,101,21)(4,110,30,102,22)(5,111,31,103,23)(6,112,32,104,24)(7,105,25,97,17)(8,106,26,98,18)(9,46,118,38,90)(10,47,119,39,91)(11,48,120,40,92)(12,41,113,33,93)(13,42,114,34,94)(14,43,115,35,95)(15,44,116,36,96)(16,45,117,37,89)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,165,153,73,145)(66,166,154,74,146)(67,167,155,75,147)(68,168,156,76,148)(69,161,157,77,149)(70,162,158,78,150)(71,163,159,79,151)(72,164,160,80,152)(81,318,234,310,226)(82,319,235,311,227)(83,320,236,312,228)(84,313,237,305,229)(85,314,238,306,230)(86,315,239,307,231)(87,316,240,308,232)(88,317,233,309,225)(169,245,261,177,253)(170,246,262,178,254)(171,247,263,179,255)(172,248,264,180,256)(173,241,257,181,249)(174,242,258,182,250)(175,243,259,183,251)(176,244,260,184,252)(185,201,277,193,269)(186,202,278,194,270)(187,203,279,195,271)(188,204,280,196,272)(189,205,273,197,265)(190,206,274,198,266)(191,207,275,199,267)(192,208,276,200,268)(209,285,301,217,293)(210,286,302,218,294)(211,287,303,219,295)(212,288,304,220,296)(213,281,297,221,289)(214,282,298,222,290)(215,283,299,223,291)(216,284,300,224,292), (1,68,125,34)(2,65,126,39)(3,70,127,36)(4,67,128,33)(5,72,121,38)(6,69,122,35)(7,66,123,40)(8,71,124,37)(9,31,160,57)(10,28,153,62)(11,25,154,59)(12,30,155,64)(13,27,156,61)(14,32,157,58)(15,29,158,63)(16,26,159,60)(17,146,51,120)(18,151,52,117)(19,148,53,114)(20,145,54,119)(21,150,55,116)(22,147,56,113)(23,152,49,118)(24,149,50,115)(41,102,75,136)(42,99,76,133)(43,104,77,130)(44,101,78,135)(45,98,79,132)(46,103,80,129)(47,100,73,134)(48,97,74,131)(81,297,207,262)(82,302,208,259)(83,299,201,264)(84,304,202,261)(85,301,203,258)(86,298,204,263)(87,303,205,260)(88,300,206,257)(89,106,163,140)(90,111,164,137)(91,108,165,142)(92,105,166,139)(93,110,167,144)(94,107,168,141)(95,112,161,138)(96,109,162,143)(169,305,212,270)(170,310,213,267)(171,307,214,272)(172,312,215,269)(173,309,216,266)(174,306,209,271)(175,311,210,268)(176,308,211,265)(177,313,220,278)(178,318,221,275)(179,315,222,280)(180,320,223,277)(181,317,224,274)(182,314,217,279)(183,319,218,276)(184,316,219,273)(185,248,228,283)(186,245,229,288)(187,242,230,285)(188,247,231,282)(189,244,232,287)(190,241,225,284)(191,246,226,281)(192,243,227,286)(193,256,236,291)(194,253,237,296)(195,250,238,293)(196,255,239,290)(197,252,240,295)(198,249,233,292)(199,254,234,289)(200,251,235,294), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,283,5,287)(2,243,6,247)(3,281,7,285)(4,241,8,245)(9,273,13,277)(10,319,14,315)(11,279,15,275)(12,317,16,313)(17,209,21,213)(18,169,22,173)(19,215,23,211)(20,175,24,171)(25,217,29,221)(26,177,30,181)(27,223,31,219)(28,183,32,179)(33,225,37,229)(34,185,38,189)(35,231,39,227)(36,191,40,187)(41,233,45,237)(42,193,46,197)(43,239,47,235)(44,199,48,195)(49,176,53,172)(50,214,54,210)(51,174,55,170)(52,212,56,216)(57,184,61,180)(58,222,62,218)(59,182,63,178)(60,220,64,224)(65,192,69,188)(66,230,70,226)(67,190,71,186)(68,228,72,232)(73,200,77,196)(74,238,78,234)(75,198,79,194)(76,236,80,240)(81,166,85,162)(82,95,86,91)(83,164,87,168)(84,93,88,89)(90,205,94,201)(92,203,96,207)(97,293,101,289)(98,253,102,249)(99,291,103,295)(100,251,104,255)(105,301,109,297)(106,261,110,257)(107,299,111,303)(108,259,112,263)(113,309,117,305)(114,269,118,265)(115,307,119,311)(116,267,120,271)(121,244,125,248)(122,282,126,286)(123,242,127,246)(124,288,128,284)(129,252,133,256)(130,290,134,294)(131,250,135,254)(132,296,136,292)(137,260,141,264)(138,298,142,302)(139,258,143,262)(140,304,144,300)(145,268,149,272)(146,306,150,310)(147,266,151,270)(148,312,152,308)(153,276,157,280)(154,314,158,318)(155,274,159,278)(156,320,160,316)(161,204,165,208)(163,202,167,206), (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) );
G=PermutationGroup([[(1,107,27,99,19),(2,108,28,100,20),(3,109,29,101,21),(4,110,30,102,22),(5,111,31,103,23),(6,112,32,104,24),(7,105,25,97,17),(8,106,26,98,18),(9,46,118,38,90),(10,47,119,39,91),(11,48,120,40,92),(12,41,113,33,93),(13,42,114,34,94),(14,43,115,35,95),(15,44,116,36,96),(16,45,117,37,89),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,165,153,73,145),(66,166,154,74,146),(67,167,155,75,147),(68,168,156,76,148),(69,161,157,77,149),(70,162,158,78,150),(71,163,159,79,151),(72,164,160,80,152),(81,318,234,310,226),(82,319,235,311,227),(83,320,236,312,228),(84,313,237,305,229),(85,314,238,306,230),(86,315,239,307,231),(87,316,240,308,232),(88,317,233,309,225),(169,245,261,177,253),(170,246,262,178,254),(171,247,263,179,255),(172,248,264,180,256),(173,241,257,181,249),(174,242,258,182,250),(175,243,259,183,251),(176,244,260,184,252),(185,201,277,193,269),(186,202,278,194,270),(187,203,279,195,271),(188,204,280,196,272),(189,205,273,197,265),(190,206,274,198,266),(191,207,275,199,267),(192,208,276,200,268),(209,285,301,217,293),(210,286,302,218,294),(211,287,303,219,295),(212,288,304,220,296),(213,281,297,221,289),(214,282,298,222,290),(215,283,299,223,291),(216,284,300,224,292)], [(1,68,125,34),(2,65,126,39),(3,70,127,36),(4,67,128,33),(5,72,121,38),(6,69,122,35),(7,66,123,40),(8,71,124,37),(9,31,160,57),(10,28,153,62),(11,25,154,59),(12,30,155,64),(13,27,156,61),(14,32,157,58),(15,29,158,63),(16,26,159,60),(17,146,51,120),(18,151,52,117),(19,148,53,114),(20,145,54,119),(21,150,55,116),(22,147,56,113),(23,152,49,118),(24,149,50,115),(41,102,75,136),(42,99,76,133),(43,104,77,130),(44,101,78,135),(45,98,79,132),(46,103,80,129),(47,100,73,134),(48,97,74,131),(81,297,207,262),(82,302,208,259),(83,299,201,264),(84,304,202,261),(85,301,203,258),(86,298,204,263),(87,303,205,260),(88,300,206,257),(89,106,163,140),(90,111,164,137),(91,108,165,142),(92,105,166,139),(93,110,167,144),(94,107,168,141),(95,112,161,138),(96,109,162,143),(169,305,212,270),(170,310,213,267),(171,307,214,272),(172,312,215,269),(173,309,216,266),(174,306,209,271),(175,311,210,268),(176,308,211,265),(177,313,220,278),(178,318,221,275),(179,315,222,280),(180,320,223,277),(181,317,224,274),(182,314,217,279),(183,319,218,276),(184,316,219,273),(185,248,228,283),(186,245,229,288),(187,242,230,285),(188,247,231,282),(189,244,232,287),(190,241,225,284),(191,246,226,281),(192,243,227,286),(193,256,236,291),(194,253,237,296),(195,250,238,293),(196,255,239,290),(197,252,240,295),(198,249,233,292),(199,254,234,289),(200,251,235,294)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,195,197,199),(194,196,198,200),(201,203,205,207),(202,204,206,208),(209,211,213,215),(210,212,214,216),(217,219,221,223),(218,220,222,224),(225,227,229,231),(226,228,230,232),(233,235,237,239),(234,236,238,240),(241,243,245,247),(242,244,246,248),(249,251,253,255),(250,252,254,256),(257,259,261,263),(258,260,262,264),(265,267,269,271),(266,268,270,272),(273,275,277,279),(274,276,278,280),(281,283,285,287),(282,284,286,288),(289,291,293,295),(290,292,294,296),(297,299,301,303),(298,300,302,304),(305,307,309,311),(306,308,310,312),(313,315,317,319),(314,316,318,320)], [(1,283,5,287),(2,243,6,247),(3,281,7,285),(4,241,8,245),(9,273,13,277),(10,319,14,315),(11,279,15,275),(12,317,16,313),(17,209,21,213),(18,169,22,173),(19,215,23,211),(20,175,24,171),(25,217,29,221),(26,177,30,181),(27,223,31,219),(28,183,32,179),(33,225,37,229),(34,185,38,189),(35,231,39,227),(36,191,40,187),(41,233,45,237),(42,193,46,197),(43,239,47,235),(44,199,48,195),(49,176,53,172),(50,214,54,210),(51,174,55,170),(52,212,56,216),(57,184,61,180),(58,222,62,218),(59,182,63,178),(60,220,64,224),(65,192,69,188),(66,230,70,226),(67,190,71,186),(68,228,72,232),(73,200,77,196),(74,238,78,234),(75,198,79,194),(76,236,80,240),(81,166,85,162),(82,95,86,91),(83,164,87,168),(84,93,88,89),(90,205,94,201),(92,203,96,207),(97,293,101,289),(98,253,102,249),(99,291,103,295),(100,251,104,255),(105,301,109,297),(106,261,110,257),(107,299,111,303),(108,259,112,263),(113,309,117,305),(114,269,118,265),(115,307,119,311),(116,267,120,271),(121,244,125,248),(122,282,126,286),(123,242,127,246),(124,288,128,284),(129,252,133,256),(130,290,134,294),(131,250,135,254),(132,296,136,292),(137,260,141,264),(138,298,142,302),(139,258,143,262),(140,304,144,300),(145,268,149,272),(146,306,150,310),(147,266,151,270),(148,312,152,308),(153,276,157,280),(154,314,158,318),(155,274,159,278),(156,320,160,316),(161,204,165,208),(163,202,167,206)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×C42.30C22 | C5×C8⋊C4 | C5×Q8⋊C4 | C5×C42.C2 | C5×C4⋊Q8 | C42.30C22 | C8⋊C4 | Q8⋊C4 | C42.C2 | C4⋊Q8 | C2×C20 | C20 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 8 |
Matrix representation of C5×C42.30C22 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 9 | 4 | 3 |
| 0 | 0 | 9 | 30 | 30 | 27 |
| 0 | 0 | 11 | 37 | 22 | 9 |
| 0 | 0 | 27 | 3 | 32 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 38 | 9 | 0 |
| 0 | 0 | 30 | 4 | 19 | 32 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 32 | 7 |
| 0 | 0 | 8 | 8 | 11 | 19 |
| 0 | 0 | 26 | 13 | 17 | 16 |
| 0 | 0 | 33 | 34 | 26 | 10 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 11 | 37 | 22 | 9 |
| 0 | 0 | 30 | 9 | 4 | 3 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,30,9,11,27,0,0,9,30,37,3,0,0,4,30,22,32,0,0,3,27,9,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,14,30,0,0,40,0,38,4,0,0,0,0,9,19,0,0,0,0,0,32],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,6,8,26,33,0,0,6,8,13,34,0,0,32,11,17,26,0,0,7,19,16,10],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,11,30,0,0,0,0,37,9,40,0,0,0,22,4,0,0,0,1,9,3,0] >;
C5×C42.30C22 in GAP, Magma, Sage, TeX
C_5\times C_4^2._{30}C_2^2 % in TeX
G:=Group("C5xC4^2.30C2^2"); // GroupNames label
G:=SmallGroup(320,992);
// by ID
G=gap.SmallGroup(320,992);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1688,1766,1731,226,7004,172,10085,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^4=1,d^2=c^2,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,e*b*e^-1=b*c^2,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;
// generators/relations