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G = C8×C5⋊C8order 320 = 26·5

Direct product of C8 and C5⋊C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×C5⋊C8, C5⋊C82, C403C8, C52C86C8, C2.2(C8×F5), C10.7(C4×C8), (C2×C8).22F5, (C2×C40).18C4, C20.20(C2×C8), C4.15(D5⋊C8), (C2×C10).8C42, Dic5.9(C2×C8), C22.13(C4×F5), (C8×Dic5).23C2, (C4×Dic5).350C22, C2.2(C4×C5⋊C8), (C2×C5⋊C8).3C4, C4.12(C2×C5⋊C8), (C4×C5⋊C8).10C2, (C2×C52C8).32C4, (C2×C4).150(C2×F5), (C2×C20).156(C2×C4), (C2×Dic5).119(C2×C4), SmallGroup(320,216)

Series: Derived Chief Lower central Upper central

C1C5 — C8×C5⋊C8
C1C5C10C2×C10C2×Dic5C4×Dic5C4×C5⋊C8 — C8×C5⋊C8
C5 — C8×C5⋊C8
C1C2×C8

Generators and relations for C8×C5⋊C8
 G = < a,b,c | a8=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 178 in 74 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C5, C8 [×2], C8 [×10], C2×C4, C2×C4 [×2], C10, C10 [×2], C42, C2×C8, C2×C8 [×5], Dic5 [×4], C20 [×2], C2×C10, C4×C8 [×3], C52C8 [×2], C40 [×2], C5⋊C8 [×8], C2×Dic5 [×2], C2×C20, C82, C2×C52C8, C4×Dic5, C2×C40, C2×C5⋊C8 [×4], C8×Dic5, C4×C5⋊C8 [×2], C8×C5⋊C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×12], C2×C4 [×3], C42, C2×C8 [×6], F5, C4×C8 [×3], C5⋊C8 [×2], C2×F5, C82, D5⋊C8, C4×F5, C2×C5⋊C8, C8×F5 [×2], C4×C5⋊C8, C8×C5⋊C8

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

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

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

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

80 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L 5 8A···8H8I···8AV10A10B10C20A20B20C20D40A···40H
order122244444···458···88···81010102020202040···40
size111111115···541···15···544444444···4

80 irreducible representations

dim111111111444444
type++++-+
imageC1C2C2C4C4C4C8C8C8F5C5⋊C8C2×F5D5⋊C8C4×F5C8×F5
kernelC8×C5⋊C8C8×Dic5C4×C5⋊C8C2×C52C8C2×C40C2×C5⋊C8C52C8C40C5⋊C8C2×C8C8C2×C4C4C22C2
# reps1122288832121228

Matrix representation of C8×C5⋊C8 in GL5(𝔽41)

380000
032000
003200
000320
000032
,
10000
000040
010040
001040
000140
,
30000
033113529
02740921
01322015
01226830

G:=sub<GL(5,GF(41))| [38,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[3,0,0,0,0,0,33,27,1,12,0,11,40,32,26,0,35,9,20,8,0,29,21,15,30] >;

C8×C5⋊C8 in GAP, Magma, Sage, TeX

C_8\times C_5\rtimes C_8
% in TeX

G:=Group("C8xC5:C8");
// GroupNames label

G:=SmallGroup(320,216);
// by ID

G=gap.SmallGroup(320,216);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,100,136,6278,3156]);
// Polycyclic

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

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