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G = C42.198D10order 320 = 26·5

18th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.198D10, C20.23M4(2), C4⋊C8.12D5, C52C810Q8, C57(C84Q8), C4.53(Q8×D5), C10.27(C4×Q8), (C2×C8).180D10, C4.3(C8⋊D5), C4⋊Dic5.29C4, C20.111(C2×Q8), C408C4.10C2, C10.51(C8○D4), (C4×C20).57C22, C20.8Q8.9C2, C20.303(C4○D4), (C2×C20).828C23, (C2×C40).207C22, (C2×Dic10).26C4, (C4×Dic10).10C2, C10.D4.22C4, C10.41(C2×M4(2)), C4.129(D42D5), C2.8(Dic53Q8), C2.12(D20.2C4), (C4×Dic5).204C22, (C5×C4⋊C8).18C2, (C4×C52C8).6C2, (C2×C4).71(C4×D5), C2.11(C2×C8⋊D5), C22.109(C2×C4×D5), (C2×C20).330(C2×C4), (C2×Dic5).23(C2×C4), (C2×C4).770(C22×D5), (C2×C10).184(C22×C4), (C2×C52C8).311C22, SmallGroup(320,458)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.198D10
C1C5C10C2×C10C2×C20C2×C52C8C4×C52C8 — C42.198D10
C5C2×C10 — C42.198D10
C1C2×C4C4⋊C8

Generators and relations for C42.198D10
 G = < a,b,c,d | a4=b4=1, c10=a2b, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >

Subgroups: 254 in 94 conjugacy classes, 53 normal (31 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 [×3], C2×C8 [×2], C2×C8 [×2], C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], C20, C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4×Q8, C52C8 [×2], C52C8, C40 [×2], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×3], C84Q8, C2×C52C8 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C4×C52C8, C20.8Q8 [×2], C408C4 [×2], C5×C4⋊C8, C4×Dic10, C42.198D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, M4(2) [×2], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C84Q8, C8⋊D5 [×2], C2×C4×D5, D42D5, Q8×D5, Dic53Q8, C2×C8⋊D5, D20.2C4, C42.198D10

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

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

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

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

68 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E···8L10A···10F20A···20H20I···20P40A···40P
order12224444444444445588888···810···1020···2020···2040···40
size1111111122222020202022444410···102···22···24···44···4

68 irreducible representations

dim111111111222222222444
type++++++-+++--
imageC1C2C2C2C2C2C4C4C4Q8D5M4(2)C4○D4D10D10C8○D4C4×D5C8⋊D5D42D5Q8×D5D20.2C4
kernelC42.198D10C4×C52C8C20.8Q8C408C4C5×C4⋊C8C4×Dic10C10.D4C4⋊Dic5C2×Dic10C52C8C4⋊C8C20C20C42C2×C8C10C2×C4C4C4C4C2
# reps1122114222242244816224

Matrix representation of C42.198D10 in GL4(𝔽41) generated by

1000
0100
00738
00334
,
9000
0900
00400
00040
,
10900
323200
0001
0010
,
63800
393500
0009
0090
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,7,3,0,0,38,34],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[10,32,0,0,9,32,0,0,0,0,0,1,0,0,1,0],[6,39,0,0,38,35,0,0,0,0,0,9,0,0,9,0] >;

C42.198D10 in GAP, Magma, Sage, TeX

C_4^2._{198}D_{10}
% in TeX

G:=Group("C4^2.198D10");
// GroupNames label

G:=SmallGroup(320,458);
// by ID

G=gap.SmallGroup(320,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,758,135,142,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b,d^2=a^2,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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