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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

Aliases: C5×C8⋊Q8, C407Q8, C8⋊(C5×Q8), C4.8(Q8×C10), C4⋊Q8.12C10, C20.97(C2×Q8), C8⋊C4.2C10, C4.Q8.3C10, C2.D8.8C10, (C2×C20).345D4, C10.43(C4⋊Q8), C42.31(C2×C10), C42.C2.5C10, (C2×C20).957C23, (C4×C20).273C22, (C2×C40).277C22, C22.122(D4×C10), C10.148(C8⋊C22), C10.148(C8.C22), C2.9(C5×C4⋊Q8), (C2×C4).46(C5×D4), (C5×C4⋊Q8).27C2, (C5×C4.Q8).8C2, (C5×C8⋊C4).6C2, C4⋊C4.26(C2×C10), (C2×C8).29(C2×C10), C2.23(C5×C8⋊C22), (C5×C2.D8).17C2, (C2×C10).678(C2×D4), C2.23(C5×C8.C22), (C5×C4⋊C4).246C22, (C5×C42.C2).12C2, (C2×C4).132(C22×C10), SmallGroup(320,1002)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C8⋊Q8
C1C2C4C2×C4C2×C20C5×C4⋊C4C5×C4⋊Q8 — C5×C8⋊Q8
C1C2C2×C4 — C5×C8⋊Q8
C1C2×C10C4×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 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8, C20 [×2], C20 [×6], C2×C10, C8⋊C4, C4.Q8 [×2], C2.D8 [×2], C42.C2, C4⋊Q8, C40 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C8⋊Q8, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4 [×3], C2×C40 [×2], Q8×C10, C5×C8⋊C4, C5×C4.Q8 [×2], C5×C2.D8 [×2], C5×C42.C2, C5×C4⋊Q8, C5×C8⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×4], C23, C10 [×7], C2×D4, C2×Q8 [×2], C2×C10 [×7], C4⋊Q8, C8⋊C22, C8.C22, C5×D4 [×2], C5×Q8 [×4], C22×C10, C8⋊Q8, D4×C10, Q8×C10 [×2], 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 53 16 63 23)(2 54 9 64 24)(3 55 10 57 17)(4 56 11 58 18)(5 49 12 59 19)(6 50 13 60 20)(7 51 14 61 21)(8 52 15 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)(169 207 222 180 214)(170 208 223 181 215)(171 201 224 182 216)(172 202 217 183 209)(173 203 218 184 210)(174 204 219 177 211)(175 205 220 178 212)(176 206 221 179 213)(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 96)(2 111 32 93)(3 108 25 90)(4 105 26 95)(5 110 27 92)(6 107 28 89)(7 112 29 94)(8 109 30 91)(9 160 80 142)(10 157 73 139)(11 154 74 144)(12 159 75 141)(13 156 76 138)(14 153 77 143)(15 158 78 140)(16 155 79 137)(17 149 65 131)(18 146 66 136)(19 151 67 133)(20 148 68 130)(21 145 69 135)(22 150 70 132)(23 147 71 129)(24 152 72 134)(33 98 57 116)(34 103 58 113)(35 100 59 118)(36 97 60 115)(37 102 61 120)(38 99 62 117)(39 104 63 114)(40 101 64 119)(41 176 284 246)(42 173 285 243)(43 170 286 248)(44 175 287 245)(45 172 288 242)(46 169 281 247)(47 174 282 244)(48 171 283 241)(49 165 82 128)(50 162 83 125)(51 167 84 122)(52 164 85 127)(53 161 86 124)(54 166 87 121)(55 163 88 126)(56 168 81 123)(177 289 231 307)(178 294 232 312)(179 291 225 309)(180 296 226 306)(181 293 227 311)(182 290 228 308)(183 295 229 305)(184 292 230 310)(185 268 215 250)(186 265 216 255)(187 270 209 252)(188 267 210 249)(189 272 211 254)(190 269 212 251)(191 266 213 256)(192 271 214 253)(193 276 223 258)(194 273 224 263)(195 278 217 260)(196 275 218 257)(197 280 219 262)(198 277 220 259)(199 274 221 264)(200 279 222 261)(201 298 236 316)(202 303 237 313)(203 300 238 318)(204 297 239 315)(205 302 240 320)(206 299 233 317)(207 304 234 314)(208 301 235 319)
(1 187 31 209)(2 190 32 212)(3 185 25 215)(4 188 26 210)(5 191 27 213)(6 186 28 216)(7 189 29 211)(8 192 30 214)(9 240 80 205)(10 235 73 208)(11 238 74 203)(12 233 75 206)(13 236 76 201)(14 239 77 204)(15 234 78 207)(16 237 79 202)(17 227 65 181)(18 230 66 184)(19 225 67 179)(20 228 68 182)(21 231 69 177)(22 226 70 180)(23 229 71 183)(24 232 72 178)(33 223 57 193)(34 218 58 196)(35 221 59 199)(36 224 60 194)(37 219 61 197)(38 222 62 200)(39 217 63 195)(40 220 64 198)(41 165 284 128)(42 168 285 123)(43 163 286 126)(44 166 287 121)(45 161 288 124)(46 164 281 127)(47 167 282 122)(48 162 283 125)(49 246 82 176)(50 241 83 171)(51 244 84 174)(52 247 85 169)(53 242 86 172)(54 245 87 175)(55 248 88 170)(56 243 81 173)(89 265 107 255)(90 268 108 250)(91 271 109 253)(92 266 110 256)(93 269 111 251)(94 272 112 254)(95 267 105 249)(96 270 106 252)(97 273 115 263)(98 276 116 258)(99 279 117 261)(100 274 118 264)(101 277 119 259)(102 280 120 262)(103 275 113 257)(104 278 114 260)(129 305 147 295)(130 308 148 290)(131 311 149 293)(132 306 150 296)(133 309 151 291)(134 312 152 294)(135 307 145 289)(136 310 146 292)(137 313 155 303)(138 316 156 298)(139 319 157 301)(140 314 158 304)(141 317 159 299)(142 320 160 302)(143 315 153 297)(144 318 154 300)

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

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

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

80 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B5C5D8A8B8C8D10A···10L20A···20H20I···20P20Q···20AF40A···40P
order1222444444445555888810···1020···2020···2020···2040···40
size111122448888111144441···12···24···48···84···4

80 irreducible representations

dim11111111111122224444
type++++++-++-
imageC1C2C2C2C2C2C5C10C10C10C10C10Q8D4C5×Q8C5×D4C8⋊C22C8.C22C5×C8⋊C22C5×C8.C22
kernelC5×C8⋊Q8C5×C8⋊C4C5×C4.Q8C5×C2.D8C5×C42.C2C5×C4⋊Q8C8⋊Q8C8⋊C4C4.Q8C2.D8C42.C2C4⋊Q8C40C2×C20C8C2×C4C10C10C2C2
# reps112211448844421681144

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

100000
010000
0037000
0003700
0000370
0000037
,
4000000
0400000
000010
000001
0013900
0014000
,
4020000
4010000
001713391
0014302040
001382428
0022392711
,
31220000
1100000
0019171030
007222531
0031113627
001610295

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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