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## G = C5×C8⋊3Q8order 320 = 26·5

### Direct product of C5 and C8⋊3Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8⋊3Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C4⋊Q8 — C5×C8⋊3Q8
 Lower central C1 — C2 — C2×C4 — C5×C8⋊3Q8
 Upper central C1 — C2×C10 — C4×C20 — C5×C8⋊3Q8

Generators and relations for C5×C83Q8
G = < a,b,c,d | a5=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 162 in 98 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C4×C8, C4.Q8, C4⋊Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C83Q8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C4×C40, C5×C4.Q8, C5×C4⋊Q8, C5×C83Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, SD16, C2×D4, C2×Q8, C2×C10, C4⋊Q8, C2×SD16, C5×D4, C5×Q8, C22×C10, C83Q8, C5×SD16, D4×C10, Q8×C10, C5×C4⋊Q8, C10×SD16, C5×C83Q8

Smallest permutation representation of C5×C83Q8
Regular action on 320 points
Generators in S320
(1 55 12 63 23)(2 56 13 64 24)(3 49 14 57 17)(4 50 15 58 18)(5 51 16 59 19)(6 52 9 60 20)(7 53 10 61 21)(8 54 11 62 22)(25 81 73 33 65)(26 82 74 34 66)(27 83 75 35 67)(28 84 76 36 68)(29 85 77 37 69)(30 86 78 38 70)(31 87 79 39 71)(32 88 80 40 72)(41 317 275 309 267)(42 318 276 310 268)(43 319 277 311 269)(44 320 278 312 270)(45 313 279 305 271)(46 314 280 306 272)(47 315 273 307 265)(48 316 274 308 266)(89 126 138 98 130)(90 127 139 99 131)(91 128 140 100 132)(92 121 141 101 133)(93 122 142 102 134)(94 123 143 103 135)(95 124 144 104 136)(96 125 137 97 129)(105 167 153 113 145)(106 168 154 114 146)(107 161 155 115 147)(108 162 156 116 148)(109 163 157 117 149)(110 164 158 118 150)(111 165 159 119 151)(112 166 160 120 152)(169 221 179 210 207)(170 222 180 211 208)(171 223 181 212 201)(172 224 182 213 202)(173 217 183 214 203)(174 218 184 215 204)(175 219 177 216 205)(176 220 178 209 206)(185 241 235 193 227)(186 242 236 194 228)(187 243 237 195 229)(188 244 238 196 230)(189 245 239 197 231)(190 246 240 198 232)(191 247 233 199 225)(192 248 234 200 226)(249 301 259 290 287)(250 302 260 291 288)(251 303 261 292 281)(252 304 262 293 282)(253 297 263 294 283)(254 298 264 295 284)(255 299 257 296 285)(256 300 258 289 286)
(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)
(1 111 31 135)(2 112 32 136)(3 105 25 129)(4 106 26 130)(5 107 27 131)(6 108 28 132)(7 109 29 133)(8 110 30 134)(9 156 76 128)(10 157 77 121)(11 158 78 122)(12 159 79 123)(13 160 80 124)(14 153 73 125)(15 154 74 126)(16 155 75 127)(17 145 65 97)(18 146 66 98)(19 147 67 99)(20 148 68 100)(21 149 69 101)(22 150 70 102)(23 151 71 103)(24 152 72 104)(33 137 57 113)(34 138 58 114)(35 139 59 115)(36 140 60 116)(37 141 61 117)(38 142 62 118)(39 143 63 119)(40 144 64 120)(41 212 292 243)(42 213 293 244)(43 214 294 245)(44 215 295 246)(45 216 296 247)(46 209 289 248)(47 210 290 241)(48 211 291 242)(49 167 81 96)(50 168 82 89)(51 161 83 90)(52 162 84 91)(53 163 85 92)(54 164 86 93)(55 165 87 94)(56 166 88 95)(169 249 193 273)(170 250 194 274)(171 251 195 275)(172 252 196 276)(173 253 197 277)(174 254 198 278)(175 255 199 279)(176 256 200 280)(177 257 191 271)(178 258 192 272)(179 259 185 265)(180 260 186 266)(181 261 187 267)(182 262 188 268)(183 263 189 269)(184 264 190 270)(201 281 237 317)(202 282 238 318)(203 283 239 319)(204 284 240 320)(205 285 233 313)(206 286 234 314)(207 287 235 315)(208 288 236 316)(217 297 231 311)(218 298 232 312)(219 299 225 305)(220 300 226 306)(221 301 227 307)(222 302 228 308)(223 303 229 309)(224 304 230 310)
(1 187 31 181)(2 190 32 184)(3 185 25 179)(4 188 26 182)(5 191 27 177)(6 186 28 180)(7 189 29 183)(8 192 30 178)(9 236 76 208)(10 239 77 203)(11 234 78 206)(12 237 79 201)(13 240 80 204)(14 235 73 207)(15 238 74 202)(16 233 75 205)(17 227 65 221)(18 230 66 224)(19 225 67 219)(20 228 68 222)(21 231 69 217)(22 226 70 220)(23 229 71 223)(24 232 72 218)(33 169 57 193)(34 172 58 196)(35 175 59 199)(36 170 60 194)(37 173 61 197)(38 176 62 200)(39 171 63 195)(40 174 64 198)(41 165 292 94)(42 168 293 89)(43 163 294 92)(44 166 295 95)(45 161 296 90)(46 164 289 93)(47 167 290 96)(48 162 291 91)(49 241 81 210)(50 244 82 213)(51 247 83 216)(52 242 84 211)(53 245 85 214)(54 248 86 209)(55 243 87 212)(56 246 88 215)(97 307 145 301)(98 310 146 304)(99 305 147 299)(100 308 148 302)(101 311 149 297)(102 306 150 300)(103 309 151 303)(104 312 152 298)(105 259 129 265)(106 262 130 268)(107 257 131 271)(108 260 132 266)(109 263 133 269)(110 258 134 272)(111 261 135 267)(112 264 136 270)(113 249 137 273)(114 252 138 276)(115 255 139 279)(116 250 140 274)(117 253 141 277)(118 256 142 280)(119 251 143 275)(120 254 144 278)(121 319 157 283)(122 314 158 286)(123 317 159 281)(124 320 160 284)(125 315 153 287)(126 318 154 282)(127 313 155 285)(128 316 156 288)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 20A ··· 20X 20Y ··· 20AN 40A ··· 40AF order 1 2 2 2 4 ··· 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 ··· 2 8 8 8 8 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C2 C5 C10 C10 C10 Q8 D4 SD16 C5×Q8 C5×D4 C5×SD16 kernel C5×C8⋊3Q8 C4×C40 C5×C4.Q8 C5×C4⋊Q8 C8⋊3Q8 C4×C8 C4.Q8 C4⋊Q8 C40 C2×C20 C20 C8 C2×C4 C4 # reps 1 1 4 2 4 4 16 8 4 2 8 16 8 32

Matrix representation of C5×C83Q8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 40 0 0 0 0 40 0 0 0 0 15 26 0 0 15 15
,
 3 2 0 0 36 38 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 27 9 0 0 0 0 24 32 0 0 32 17
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,15,15,0,0,26,15],[3,36,0,0,2,38,0,0,0,0,40,0,0,0,0,40],[32,27,0,0,0,9,0,0,0,0,24,32,0,0,32,17] >;

C5×C83Q8 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_3Q_8
% in TeX

G:=Group("C5xC8:3Q8");
// GroupNames label

G:=SmallGroup(320,999);
// by ID

G=gap.SmallGroup(320,999);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,436,10085,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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