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## G = C5×C4.Q16order 320 = 26·5

### Direct product of C5 and C4.Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C4.Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C4⋊Q8 — C5×C4.Q16
 Lower central C1 — C2 — C2×C4 — C5×C4.Q16
 Upper central C1 — C2×C10 — C4×C20 — C5×C4.Q16

Generators and relations for C5×C4.Q16
G = < a,b,c,d | a5=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >

Subgroups: 154 in 96 conjugacy classes, 58 normal (34 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], C10 [×3], C42, C42, C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×Q8, C2×Q8, C20 [×2], C20 [×2], C20 [×6], C2×C10, Q8⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8, C40 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C5×Q8 [×3], C4.Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], Q8×C10, Q8×C10, C5×Q8⋊C4 [×2], C5×C4⋊C8, C5×C2.D8 [×2], Q8×C20, C5×C4⋊Q8, C5×C4.Q16
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×2], C23, C10 [×7], Q16 [×2], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], C22⋊Q8, C2×Q16, C8⋊C22, C5×D4 [×2], C5×Q8 [×2], C22×C10, C4.Q16, C5×Q16 [×2], D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C10×Q16, C5×C8⋊C22, C5×C4.Q16

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4I 4J 4K 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 20A ··· 20P 20Q ··· 20AJ 20AK ··· 20AR 40A ··· 40P order 1 2 2 2 4 4 4 4 4 ··· 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 4 ··· 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 Q8 Q16 C4○D4 C5×D4 C5×Q8 C5×Q16 C5×C4○D4 C8⋊C22 C5×C8⋊C22 kernel C5×C4.Q16 C5×Q8⋊C4 C5×C4⋊C8 C5×C2.D8 Q8×C20 C5×C4⋊Q8 C4.Q16 Q8⋊C4 C4⋊C8 C2.D8 C4×Q8 C4⋊Q8 C2×C20 C5×Q8 C20 C20 C2×C4 Q8 C4 C4 C10 C2 # reps 1 2 1 2 1 1 4 8 4 8 4 4 2 2 4 2 8 8 16 8 1 4

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

 10 0 0 0 0 10 0 0 0 0 18 0 0 0 0 18
,
 32 18 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 9 23 0 0 9 32 0 0 0 0 12 29 0 0 12 12
,
 1 39 0 0 0 40 0 0 0 0 40 11 0 0 11 1
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[32,0,0,0,18,9,0,0,0,0,1,0,0,0,0,1],[9,9,0,0,23,32,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,39,40,0,0,0,0,40,11,0,0,11,1] >;

C5×C4.Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C5xC4.Q16");
// GroupNames label

G:=SmallGroup(320,978);
// by ID

G=gap.SmallGroup(320,978);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1400,589,288,1766,856,10085,2539,124]);
// Polycyclic

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

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