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## G = C5×C8⋊Q8order 320 = 26·5

### Direct product of C5 and C8⋊Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C8⋊Q8
G = < a,b,c,d | a5=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-1 >

Subgroups: 146 in 90 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C8⋊Q8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C5×C8⋊C4, C5×C4.Q8, C5×C2.D8, C5×C42.C2, C5×C4⋊Q8, C5×C8⋊Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C2×C10, C4⋊Q8, C8⋊C22, C8.C22, C5×D4, C5×Q8, C22×C10, C8⋊Q8, D4×C10, Q8×C10, C5×C4⋊Q8, C5×C8⋊C22, C5×C8.C22, C5×C8⋊Q8

Smallest permutation representation of C5×C8⋊Q8
Regular action on 320 points
Generators in S320
(1 14 55 63 23)(2 15 56 64 24)(3 16 49 57 17)(4 9 50 58 18)(5 10 51 59 19)(6 11 52 60 20)(7 12 53 61 21)(8 13 54 62 22)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 317 274 309 266)(42 318 275 310 267)(43 319 276 311 268)(44 320 277 312 269)(45 313 278 305 270)(46 314 279 306 271)(47 315 280 307 272)(48 316 273 308 265)(89 125 138 97 130)(90 126 139 98 131)(91 127 140 99 132)(92 128 141 100 133)(93 121 142 101 134)(94 122 143 102 135)(95 123 144 103 136)(96 124 137 104 129)(105 168 154 113 146)(106 161 155 114 147)(107 162 156 115 148)(108 163 157 116 149)(109 164 158 117 150)(110 165 159 118 151)(111 166 160 119 152)(112 167 153 120 145)(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)(185 248 235 193 227)(186 241 236 194 228)(187 242 237 195 229)(188 243 238 196 230)(189 244 239 197 231)(190 245 240 198 232)(191 246 233 199 225)(192 247 234 200 226)(249 285 300 257 292)(250 286 301 258 293)(251 287 302 259 294)(252 288 303 260 295)(253 281 304 261 296)(254 282 297 262 289)(255 283 298 263 290)(256 284 299 264 291)
(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)
(1 106 31 94)(2 111 32 91)(3 108 25 96)(4 105 26 93)(5 110 27 90)(6 107 28 95)(7 112 29 92)(8 109 30 89)(9 168 81 121)(10 165 82 126)(11 162 83 123)(12 167 84 128)(13 164 85 125)(14 161 86 122)(15 166 87 127)(16 163 88 124)(17 149 65 129)(18 146 66 134)(19 151 67 131)(20 148 68 136)(21 145 69 133)(22 150 70 130)(23 147 71 135)(24 152 72 132)(33 104 57 116)(34 101 58 113)(35 98 59 118)(36 103 60 115)(37 100 61 120)(38 97 62 117)(39 102 63 114)(40 99 64 119)(41 216 284 246)(42 213 285 243)(43 210 286 248)(44 215 287 245)(45 212 288 242)(46 209 281 247)(47 214 282 244)(48 211 283 241)(49 157 73 137)(50 154 74 142)(51 159 75 139)(52 156 76 144)(53 153 77 141)(54 158 78 138)(55 155 79 143)(56 160 80 140)(169 258 193 276)(170 263 194 273)(171 260 195 278)(172 257 196 275)(173 262 197 280)(174 259 198 277)(175 264 199 274)(176 261 200 279)(177 256 191 266)(178 253 192 271)(179 250 185 268)(180 255 186 265)(181 252 187 270)(182 249 188 267)(183 254 189 272)(184 251 190 269)(201 303 237 313)(202 300 238 318)(203 297 239 315)(204 302 240 320)(205 299 233 317)(206 304 234 314)(207 301 235 319)(208 298 236 316)(217 289 231 307)(218 294 232 312)(219 291 225 309)(220 296 226 306)(221 293 227 311)(222 290 228 308)(223 295 229 305)(224 292 230 310)
(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)(9 243 81 213)(10 246 82 216)(11 241 83 211)(12 244 84 214)(13 247 85 209)(14 242 86 212)(15 245 87 215)(16 248 88 210)(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)(41 165 284 126)(42 168 285 121)(43 163 286 124)(44 166 287 127)(45 161 288 122)(46 164 281 125)(47 167 282 128)(48 162 283 123)(49 235 73 207)(50 238 74 202)(51 233 75 205)(52 236 76 208)(53 239 77 203)(54 234 78 206)(55 237 79 201)(56 240 80 204)(89 271 109 253)(90 266 110 256)(91 269 111 251)(92 272 112 254)(93 267 105 249)(94 270 106 252)(95 265 107 255)(96 268 108 250)(97 279 117 261)(98 274 118 264)(99 277 119 259)(100 280 120 262)(101 275 113 257)(102 278 114 260)(103 273 115 263)(104 276 116 258)(129 311 149 293)(130 306 150 296)(131 309 151 291)(132 312 152 294)(133 307 145 289)(134 310 146 292)(135 305 147 295)(136 308 148 290)(137 319 157 301)(138 314 158 304)(139 317 159 299)(140 320 160 302)(141 315 153 297)(142 318 154 300)(143 313 155 303)(144 316 156 298)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 20A ··· 20H 20I ··· 20P 20Q ··· 20AF 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 4 4 8 8 8 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 Q8 D4 C5×Q8 C5×D4 C8⋊C22 C8.C22 C5×C8⋊C22 C5×C8.C22 kernel C5×C8⋊Q8 C5×C8⋊C4 C5×C4.Q8 C5×C2.D8 C5×C42.C2 C5×C4⋊Q8 C8⋊Q8 C8⋊C4 C4.Q8 C2.D8 C42.C2 C4⋊Q8 C40 C2×C20 C8 C2×C4 C10 C10 C2 C2 # reps 1 1 2 2 1 1 4 4 8 8 4 4 4 2 16 8 1 1 4 4

Matrix representation of C5×C8⋊Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 39 0 0 0 0 1 40 0 0
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 17 13 39 1 0 0 14 30 20 40 0 0 1 38 24 28 0 0 22 39 27 11
,
 31 22 0 0 0 0 1 10 0 0 0 0 0 0 19 17 10 30 0 0 7 22 25 31 0 0 31 11 36 27 0 0 16 10 29 5

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,1,0,0,0,0,0,0,1,0,0],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,17,14,1,22,0,0,13,30,38,39,0,0,39,20,24,27,0,0,1,40,28,11],[31,1,0,0,0,0,22,10,0,0,0,0,0,0,19,7,31,16,0,0,17,22,11,10,0,0,10,25,36,29,0,0,30,31,27,5] >;

C5×C8⋊Q8 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes Q_8
% in TeX

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

G:=SmallGroup(320,1002);
// by ID

G=gap.SmallGroup(320,1002);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,1731,436,10085,124]);
// 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,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

׿
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