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G = C42.68D10order 320 = 26·5

68th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.68D10, C54(C8⋊Q8), C52C83Q8, C4.33(Q8×D5), C4⋊C4.74D10, (C2×C20).84D4, C20.33(C2×Q8), C10.29(C4⋊Q8), C42.C2.4D5, C202Q8.17C2, (C4×C20).113C22, (C2×C20).383C23, C10.D8.14C2, C20.Q8.15C2, C2.20(D4⋊D10), C42.D5.5C2, C10.121(C8⋊C22), C2.9(Dic5⋊Q8), C4⋊Dic5.153C22, C2.21(D4.9D10), C10.122(C8.C22), (C2×C10).514(C2×D4), (C2×C4).65(C5⋊D4), (C5×C42.C2).3C2, (C5×C4⋊C4).121C22, (C2×C4).481(C22×D5), C22.187(C2×C5⋊D4), (C2×C52C8).125C22, SmallGroup(320,692)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.68D10
C1C5C10C2×C10C2×C20C2×C52C8C42.D5 — C42.68D10
C5C10C2×C20 — C42.68D10
C1C22C42C42.C2

Generators and relations for C42.68D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc9 >

Subgroups: 302 in 90 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×4], C2×C10, C8⋊C4, C4.Q8 [×2], C2.D8 [×2], C42.C2, C4⋊Q8, C52C8 [×4], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C8⋊Q8, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C42.D5, C10.D8 [×2], C20.Q8 [×2], C202Q8, C5×C42.C2, C42.68D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C8⋊C22, C8.C22, C5⋊D4 [×2], C22×D5, C8⋊Q8, Q8×D5 [×2], C2×C5⋊D4, Dic5⋊Q8, D4⋊D10, D4.9D10, C42.68D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12224444444455888810···1020···2020···20
size1111224488404022202020202···24···48···8

44 irreducible representations

dim11111122222244444
type++++++-+++++--+-
imageC1C2C2C2C2C2Q8D4D5D10D10C5⋊D4C8⋊C22C8.C22Q8×D5D4⋊D10D4.9D10
kernelC42.68D10C42.D5C10.D8C20.Q8C202Q8C5×C42.C2C52C8C2×C20C42.C2C42C4⋊C4C2×C4C10C10C4C2C2
# reps11221142224811444

Matrix representation of C42.68D10 in GL6(𝔽41)

090000
900000
002133715
002839264
002133928
002839132
,
4000000
0400000
0010390
0001039
0010400
0001040
,
0400000
100000
0028132035
002824625
0038101328
0031161317
,
3200000
090000
005300
00333600
00533638
00333685

G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,2,28,2,28,0,0,13,39,13,39,0,0,37,26,39,13,0,0,15,4,28,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,28,28,38,31,0,0,13,24,10,16,0,0,20,6,13,13,0,0,35,25,28,17],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,5,33,5,33,0,0,3,36,3,36,0,0,0,0,36,8,0,0,0,0,38,5] >;

C42.68D10 in GAP, Magma, Sage, TeX

C_4^2._{68}D_{10}
% in TeX

G:=Group("C4^2.68D10");
// GroupNames label

G:=SmallGroup(320,692);
// by ID

G=gap.SmallGroup(320,692);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,422,471,58,438,102,12550]);
// Polycyclic

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

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