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G = C42.210D10order 320 = 26·5

30th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.210D10, C20.25M4(2), C52C811Q8, C58(C84Q8), (C4×Q8).5D5, C4.58(Q8×D5), C4⋊C4.9Dic5, (Q8×C20).6C2, C10.36(C4×Q8), C2.5(Q8×Dic5), (Q8×C10).19C4, C20.116(C2×Q8), (C2×Q8).5Dic5, C203C8.18C2, C10.66(C8○D4), (C4×C20).95C22, C4.3(C4.Dic5), C20.339(C4○D4), (C2×C20).852C23, C4.59(Q82D5), C10.76(C2×M4(2)), C2.8(D4.Dic5), C42.D5.3C2, C22.47(C22×Dic5), (C5×C4⋊C4).27C4, (C4×C52C8).8C2, (C2×C20).339(C2×C4), (C2×C4).45(C2×Dic5), C2.10(C2×C4.Dic5), (C2×C4).794(C22×D5), (C2×C10).290(C22×C4), (C2×C52C8).206C22, SmallGroup(320,651)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.210D10
C1C5C10C20C2×C20C2×C52C8C4×C52C8 — C42.210D10
C5C2×C10 — C42.210D10
C1C2×C4C4×Q8

Generators and relations for C42.210D10
 G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >

Subgroups: 190 in 94 conjugacy classes, 61 normal (33 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C42 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×Q8, C20 [×2], C20 [×2], C20 [×5], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8 [×3], C4×Q8, C52C8 [×2], C52C8 [×3], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C84Q8, C2×C52C8 [×2], C2×C52C8 [×2], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], Q8×C10, C4×C52C8, C42.D5 [×2], C203C8, C203C8 [×2], Q8×C20, C42.210D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, M4(2) [×2], C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], C4×Q8, C2×M4(2), C8○D4, C2×Dic5 [×6], C22×D5, C84Q8, C4.Dic5 [×2], Q8×D5, Q82D5, C22×Dic5, C2×C4.Dic5, Q8×Dic5, D4.Dic5, C42.210D10

Smallest permutation representation of C42.210D10
Regular action on 320 points
Generators in S320
(1 126 11 136)(2 137 12 127)(3 128 13 138)(4 139 14 129)(5 130 15 140)(6 121 16 131)(7 132 17 122)(8 123 18 133)(9 134 19 124)(10 125 20 135)(21 196 31 186)(22 187 32 197)(23 198 33 188)(24 189 34 199)(25 200 35 190)(26 191 36 181)(27 182 37 192)(28 193 38 183)(29 184 39 194)(30 195 40 185)(41 234 51 224)(42 225 52 235)(43 236 53 226)(44 227 54 237)(45 238 55 228)(46 229 56 239)(47 240 57 230)(48 231 58 221)(49 222 59 232)(50 233 60 223)(61 147 71 157)(62 158 72 148)(63 149 73 159)(64 160 74 150)(65 151 75 141)(66 142 76 152)(67 153 77 143)(68 144 78 154)(69 155 79 145)(70 146 80 156)(81 317 91 307)(82 308 92 318)(83 319 93 309)(84 310 94 320)(85 301 95 311)(86 312 96 302)(87 303 97 313)(88 314 98 304)(89 305 99 315)(90 316 100 306)(101 244 111 254)(102 255 112 245)(103 246 113 256)(104 257 114 247)(105 248 115 258)(106 259 116 249)(107 250 117 260)(108 241 118 251)(109 252 119 242)(110 243 120 253)(161 276 171 266)(162 267 172 277)(163 278 173 268)(164 269 174 279)(165 280 175 270)(166 271 176 261)(167 262 177 272)(168 273 178 263)(169 264 179 274)(170 275 180 265)(201 296 211 286)(202 287 212 297)(203 298 213 288)(204 289 214 299)(205 300 215 290)(206 291 216 281)(207 282 217 292)(208 293 218 283)(209 284 219 294)(210 295 220 285)
(1 170 120 232)(2 171 101 233)(3 172 102 234)(4 173 103 235)(5 174 104 236)(6 175 105 237)(7 176 106 238)(8 177 107 239)(9 178 108 240)(10 179 109 221)(11 180 110 222)(12 161 111 223)(13 162 112 224)(14 163 113 225)(15 164 114 226)(16 165 115 227)(17 166 116 228)(18 167 117 229)(19 168 118 230)(20 169 119 231)(21 68 89 287)(22 69 90 288)(23 70 91 289)(24 71 92 290)(25 72 93 291)(26 73 94 292)(27 74 95 293)(28 75 96 294)(29 76 97 295)(30 77 98 296)(31 78 99 297)(32 79 100 298)(33 80 81 299)(34 61 82 300)(35 62 83 281)(36 63 84 282)(37 64 85 283)(38 65 86 284)(39 66 87 285)(40 67 88 286)(41 138 267 245)(42 139 268 246)(43 140 269 247)(44 121 270 248)(45 122 271 249)(46 123 272 250)(47 124 273 251)(48 125 274 252)(49 126 275 253)(50 127 276 254)(51 128 277 255)(52 129 278 256)(53 130 279 257)(54 131 280 258)(55 132 261 259)(56 133 262 260)(57 134 263 241)(58 135 264 242)(59 136 265 243)(60 137 266 244)(141 302 209 193)(142 303 210 194)(143 304 211 195)(144 305 212 196)(145 306 213 197)(146 307 214 198)(147 308 215 199)(148 309 216 200)(149 310 217 181)(150 311 218 182)(151 312 219 183)(152 313 220 184)(153 314 201 185)(154 315 202 186)(155 316 203 187)(156 317 204 188)(157 318 205 189)(158 319 206 190)(159 320 207 191)(160 301 208 192)
(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)
(1 152 222 194 120 220 180 303)(2 209 223 312 101 141 161 183)(3 150 224 192 102 218 162 301)(4 207 225 310 103 159 163 181)(5 148 226 190 104 216 164 319)(6 205 227 308 105 157 165 199)(7 146 228 188 106 214 166 317)(8 203 229 306 107 155 167 197)(9 144 230 186 108 212 168 315)(10 201 231 304 109 153 169 195)(11 142 232 184 110 210 170 313)(12 219 233 302 111 151 171 193)(13 160 234 182 112 208 172 311)(14 217 235 320 113 149 173 191)(15 158 236 200 114 206 174 309)(16 215 237 318 115 147 175 189)(17 156 238 198 116 204 176 307)(18 213 239 316 117 145 177 187)(19 154 240 196 118 202 178 305)(20 211 221 314 119 143 179 185)(21 251 297 263 89 124 78 57)(22 133 298 46 90 260 79 272)(23 249 299 261 91 122 80 55)(24 131 300 44 92 258 61 270)(25 247 281 279 93 140 62 53)(26 129 282 42 94 256 63 268)(27 245 283 277 95 138 64 51)(28 127 284 60 96 254 65 266)(29 243 285 275 97 136 66 49)(30 125 286 58 98 252 67 264)(31 241 287 273 99 134 68 47)(32 123 288 56 100 250 69 262)(33 259 289 271 81 132 70 45)(34 121 290 54 82 248 71 280)(35 257 291 269 83 130 72 43)(36 139 292 52 84 246 73 278)(37 255 293 267 85 128 74 41)(38 137 294 50 86 244 75 276)(39 253 295 265 87 126 76 59)(40 135 296 48 88 242 77 274)

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

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

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

68 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H8I8J8K8L10A···10F20A···20H20I···20AF
order1222444444444444558···8888810···1020···2020···20
size11111111222244442210···10202020202···22···24···4

68 irreducible representations

dim1111111222222222444
type+++++-++---+
imageC1C2C2C2C2C4C4Q8D5M4(2)C4○D4D10Dic5Dic5C8○D4C4.Dic5Q8×D5Q82D5D4.Dic5
kernelC42.210D10C4×C52C8C42.D5C203C8Q8×C20C5×C4⋊C4Q8×C10C52C8C4×Q8C20C20C42C4⋊C4C2×Q8C10C4C4C4C2
# reps11231622242662416224

Matrix representation of C42.210D10 in GL4(𝔽41) generated by

40000
04000
003119
004010
,
9000
0900
0090
0009
,
212000
211800
002937
002612
,
373000
6400
001420
00527
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,31,40,0,0,19,10],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[21,21,0,0,20,18,0,0,0,0,29,26,0,0,37,12],[37,6,0,0,30,4,0,0,0,0,14,5,0,0,20,27] >;

C42.210D10 in GAP, Magma, Sage, TeX

C_4^2._{210}D_{10}
% in TeX

G:=Group("C4^2.210D10");
// GroupNames label

G:=SmallGroup(320,651);
// by ID

G=gap.SmallGroup(320,651);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,100,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^9>;
// generators/relations

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