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G = C4×C5⋊Q16order 320 = 26·5

Direct product of C4 and C5⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C5⋊Q16, C208Q16, C42.212D10, C55(C4×Q16), (C4×Q8).6D5, Q8.5(C4×D5), (Q8×C20).7C2, C4⋊C4.255D10, C10.105(C4×D4), (C2×C20).259D4, C10.35(C2×Q16), C4.42(C4○D20), C20.62(C4○D4), C10.94(C4○D8), C20.62(C22×C4), (C2×Q8).162D10, (C4×C20).100C22, (C2×C20).349C23, (C4×Dic10).14C2, Dic10.31(C2×C4), C10.D8.18C2, Q8⋊Dic5.17C2, C10.Q16.17C2, C2.6(D4.8D10), C4⋊Dic5.332C22, (Q8×C10).197C22, (C2×Dic10).274C22, C4.27(C2×C4×D5), (C4×C52C8).9C2, C2.21(C4×C5⋊D4), C2.3(C2×C5⋊Q16), C52C8.25(C2×C4), (C5×Q8).27(C2×C4), (C2×C10).480(C2×D4), (C2×C5⋊Q16).10C2, C22.81(C2×C5⋊D4), (C2×C4).104(C5⋊D4), (C5×C4⋊C4).286C22, (C2×C4).449(C22×D5), (C2×C52C8).255C22, SmallGroup(320,656)

Series: Derived Chief Lower central Upper central

C1C20 — C4×C5⋊Q16
C1C5C10C2×C10C2×C20C2×Dic10C2×C5⋊Q16 — C4×C5⋊Q16
C5C10C20 — C4×C5⋊Q16
C1C2×C4C42C4×Q8

Generators and relations for C4×C5⋊Q16
 G = < a,b,c,d | a4=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 310 in 110 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C5, C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20 [×4], C2×C10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8, C4×Q8, C2×Q16, C52C8 [×2], C52C8, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, C4×Q16, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, C5⋊Q16 [×4], C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×C52C8, C10.D8, C10.Q16, Q8⋊Dic5, C4×Dic10, C2×C5⋊Q16, Q8×C20, C4×C5⋊Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, Q16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×Q16, C4○D8, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4×Q16, C5⋊Q16 [×2], C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×C5⋊Q16, D4.8D10, C4×C5⋊Q16

Smallest permutation representation of C4×C5⋊Q16
Regular action on 320 points
Generators in S320
(1 264 50 102)(2 257 51 103)(3 258 52 104)(4 259 53 97)(5 260 54 98)(6 261 55 99)(7 262 56 100)(8 263 49 101)(9 311 187 108)(10 312 188 109)(11 305 189 110)(12 306 190 111)(13 307 191 112)(14 308 192 105)(15 309 185 106)(16 310 186 107)(17 237 222 200)(18 238 223 193)(19 239 224 194)(20 240 217 195)(21 233 218 196)(22 234 219 197)(23 235 220 198)(24 236 221 199)(25 137 153 65)(26 138 154 66)(27 139 155 67)(28 140 156 68)(29 141 157 69)(30 142 158 70)(31 143 159 71)(32 144 160 72)(33 230 151 73)(34 231 152 74)(35 232 145 75)(36 225 146 76)(37 226 147 77)(38 227 148 78)(39 228 149 79)(40 229 150 80)(41 88 161 280)(42 81 162 273)(43 82 163 274)(44 83 164 275)(45 84 165 276)(46 85 166 277)(47 86 167 278)(48 87 168 279)(57 91 172 268)(58 92 173 269)(59 93 174 270)(60 94 175 271)(61 95 176 272)(62 96 169 265)(63 89 170 266)(64 90 171 267)(113 319 301 180)(114 320 302 181)(115 313 303 182)(116 314 304 183)(117 315 297 184)(118 316 298 177)(119 317 299 178)(120 318 300 179)(121 202 282 242)(122 203 283 243)(123 204 284 244)(124 205 285 245)(125 206 286 246)(126 207 287 247)(127 208 288 248)(128 201 281 241)(129 210 295 255)(130 211 296 256)(131 212 289 249)(132 213 290 250)(133 214 291 251)(134 215 292 252)(135 216 293 253)(136 209 294 254)
(1 312 119 46 63)(2 64 47 120 305)(3 306 113 48 57)(4 58 41 114 307)(5 308 115 42 59)(6 60 43 116 309)(7 310 117 44 61)(8 62 45 118 311)(9 101 265 276 177)(10 178 277 266 102)(11 103 267 278 179)(12 180 279 268 104)(13 97 269 280 181)(14 182 273 270 98)(15 99 271 274 183)(16 184 275 272 100)(17 121 129 70 228)(18 229 71 130 122)(19 123 131 72 230)(20 231 65 132 124)(21 125 133 66 232)(22 225 67 134 126)(23 127 135 68 226)(24 227 69 136 128)(25 213 205 240 152)(26 145 233 206 214)(27 215 207 234 146)(28 147 235 208 216)(29 209 201 236 148)(30 149 237 202 210)(31 211 203 238 150)(32 151 239 204 212)(33 194 244 249 160)(34 153 250 245 195)(35 196 246 251 154)(36 155 252 247 197)(37 198 248 253 156)(38 157 254 241 199)(39 200 242 255 158)(40 159 256 243 193)(49 169 165 298 108)(50 109 299 166 170)(51 171 167 300 110)(52 111 301 168 172)(53 173 161 302 112)(54 105 303 162 174)(55 175 163 304 106)(56 107 297 164 176)(73 224 284 289 144)(74 137 290 285 217)(75 218 286 291 138)(76 139 292 287 219)(77 220 288 293 140)(78 141 294 281 221)(79 222 282 295 142)(80 143 296 283 223)(81 93 260 192 313)(82 314 185 261 94)(83 95 262 186 315)(84 316 187 263 96)(85 89 264 188 317)(86 318 189 257 90)(87 91 258 190 319)(88 320 191 259 92)
(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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 229 13 225)(10 228 14 232)(11 227 15 231)(12 226 16 230)(17 182 21 178)(18 181 22 177)(19 180 23 184)(20 179 24 183)(33 111 37 107)(34 110 38 106)(35 109 39 105)(36 108 40 112)(41 207 45 203)(42 206 46 202)(43 205 47 201)(44 204 48 208)(49 159 53 155)(50 158 54 154)(51 157 55 153)(52 156 56 160)(57 216 61 212)(58 215 62 211)(59 214 63 210)(60 213 64 209)(65 103 69 99)(66 102 70 98)(67 101 71 97)(68 100 72 104)(73 190 77 186)(74 189 78 185)(75 188 79 192)(76 187 80 191)(81 286 85 282)(82 285 86 281)(83 284 87 288)(84 283 88 287)(89 295 93 291)(90 294 94 290)(91 293 95 289)(92 292 96 296)(113 235 117 239)(114 234 118 238)(115 233 119 237)(116 240 120 236)(121 273 125 277)(122 280 126 276)(123 279 127 275)(124 278 128 274)(129 270 133 266)(130 269 134 265)(131 268 135 272)(132 267 136 271)(137 257 141 261)(138 264 142 260)(139 263 143 259)(140 262 144 258)(145 312 149 308)(146 311 150 307)(147 310 151 306)(148 309 152 305)(161 247 165 243)(162 246 166 242)(163 245 167 241)(164 244 168 248)(169 256 173 252)(170 255 174 251)(171 254 175 250)(172 253 176 249)(193 302 197 298)(194 301 198 297)(195 300 199 304)(196 299 200 303)(217 318 221 314)(218 317 222 313)(219 316 223 320)(220 315 224 319)

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

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

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

68 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B8A···8H10A···10F20A···20H20I···20AF
order12224444444444444444558···810···1020···2020···20
size1111111122224444202020202210···102···22···24···4

68 irreducible representations

dim1111111112222222222244
type++++++++++-+++-
imageC1C2C2C2C2C2C2C2C4D4D5Q16C4○D4D10D10D10C4○D8C5⋊D4C4×D5C4○D20C5⋊Q16D4.8D10
kernelC4×C5⋊Q16C4×C52C8C10.D8C10.Q16Q8⋊Dic5C4×Dic10C2×C5⋊Q16Q8×C20C5⋊Q16C2×C20C4×Q8C20C20C42C4⋊C4C2×Q8C10C2×C4Q8C4C4C2
# reps1111111182242222488844

Matrix representation of C4×C5⋊Q16 in GL4(𝔽41) generated by

9000
0900
0010
0001
,
344000
1000
0010
0001
,
91100
303200
001229
001212
,
40000
04000
001515
001526
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[9,30,0,0,11,32,0,0,0,0,12,12,0,0,29,12],[40,0,0,0,0,40,0,0,0,0,15,15,0,0,15,26] >;

C4×C5⋊Q16 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C4xC5:Q16");
// GroupNames label

G:=SmallGroup(320,656);
// by ID

G=gap.SmallGroup(320,656);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,58,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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