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G = C8×Dic10order 320 = 26·5

Direct product of C8 and Dic10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8×Dic10, C4010Q8, C42.250D10, C53(C8×Q8), C4.9(C8×D5), (C4×C8).4D5, (C4×C40).21C2, C20.53(C2×C8), C10.14(C4×Q8), C20.79(C2×Q8), (C2×C8).337D10, C4⋊Dic5.34C4, Dic5.3(C2×C8), C203C8.24C2, C2.1(C4×Dic10), C10.25(C8○D4), C10.25(C22×C8), (C8×Dic5).14C2, C4.44(C2×Dic10), C20.238(C4○D4), C4.122(C4○D20), (C2×C40).338C22, (C4×C20).321C22, (C2×C20).800C23, (C2×Dic10).31C4, (C4×Dic10).27C2, C10.D4.28C4, C20.8Q8.16C2, C2.1(D20.3C4), (C4×Dic5).293C22, C2.4(D5×C2×C8), C22.33(C2×C4×D5), (C2×C4).101(C4×D5), (C2×C20).391(C2×C4), (C2×Dic5).90(C2×C4), (C2×C4).742(C22×D5), (C2×C10).156(C22×C4), (C2×C52C8).296C22, SmallGroup(320,305)

Series: Derived Chief Lower central Upper central

C1C10 — C8×Dic10
C1C5C10C20C2×C20C4×Dic5C4×Dic10 — C8×Dic10
C5C10 — C8×Dic10
C1C2×C8C4×C8

Generators and relations for C8×Dic10
 G = < a,b,c | a8=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 254 in 102 conjugacy classes, 63 normal (33 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C20, C2×C10, C4×C8, C4×C8 [×2], C4⋊C8 [×3], C4×Q8, C52C8 [×2], C40 [×2], C40, Dic10 [×4], C2×Dic5 [×4], C2×C20 [×3], C8×Q8, C2×C52C8 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C203C8, C8×Dic5 [×2], C20.8Q8 [×2], C4×C40, C4×Dic10, C8×Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, C22×C8, C8○D4, Dic10 [×2], C4×D5 [×2], C22×D5, C8×Q8, C8×D5 [×2], C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10, D5×C2×C8, D20.3C4, C8×Dic10

Smallest permutation representation of C8×Dic10
Regular action on 320 points
Generators in S320
(1 251 38 188 207 74 278 129)(2 252 39 189 208 75 279 130)(3 253 40 190 209 76 280 131)(4 254 21 191 210 77 261 132)(5 255 22 192 211 78 262 133)(6 256 23 193 212 79 263 134)(7 257 24 194 213 80 264 135)(8 258 25 195 214 61 265 136)(9 259 26 196 215 62 266 137)(10 260 27 197 216 63 267 138)(11 241 28 198 217 64 268 139)(12 242 29 199 218 65 269 140)(13 243 30 200 219 66 270 121)(14 244 31 181 220 67 271 122)(15 245 32 182 201 68 272 123)(16 246 33 183 202 69 273 124)(17 247 34 184 203 70 274 125)(18 248 35 185 204 71 275 126)(19 249 36 186 205 72 276 127)(20 250 37 187 206 73 277 128)(41 146 297 102 308 237 175 98)(42 147 298 103 309 238 176 99)(43 148 299 104 310 239 177 100)(44 149 300 105 311 240 178 81)(45 150 281 106 312 221 179 82)(46 151 282 107 313 222 180 83)(47 152 283 108 314 223 161 84)(48 153 284 109 315 224 162 85)(49 154 285 110 316 225 163 86)(50 155 286 111 317 226 164 87)(51 156 287 112 318 227 165 88)(52 157 288 113 319 228 166 89)(53 158 289 114 320 229 167 90)(54 159 290 115 301 230 168 91)(55 160 291 116 302 231 169 92)(56 141 292 117 303 232 170 93)(57 142 293 118 304 233 171 94)(58 143 294 119 305 234 172 95)(59 144 295 120 306 235 173 96)(60 145 296 101 307 236 174 97)
(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)
(1 164 11 174)(2 163 12 173)(3 162 13 172)(4 161 14 171)(5 180 15 170)(6 179 16 169)(7 178 17 168)(8 177 18 167)(9 176 19 166)(10 175 20 165)(21 47 31 57)(22 46 32 56)(23 45 33 55)(24 44 34 54)(25 43 35 53)(26 42 36 52)(27 41 37 51)(28 60 38 50)(29 59 39 49)(30 58 40 48)(61 104 71 114)(62 103 72 113)(63 102 73 112)(64 101 74 111)(65 120 75 110)(66 119 76 109)(67 118 77 108)(68 117 78 107)(69 116 79 106)(70 115 80 105)(81 247 91 257)(82 246 92 256)(83 245 93 255)(84 244 94 254)(85 243 95 253)(86 242 96 252)(87 241 97 251)(88 260 98 250)(89 259 99 249)(90 258 100 248)(121 234 131 224)(122 233 132 223)(123 232 133 222)(124 231 134 221)(125 230 135 240)(126 229 136 239)(127 228 137 238)(128 227 138 237)(129 226 139 236)(130 225 140 235)(141 192 151 182)(142 191 152 181)(143 190 153 200)(144 189 154 199)(145 188 155 198)(146 187 156 197)(147 186 157 196)(148 185 158 195)(149 184 159 194)(150 183 160 193)(201 292 211 282)(202 291 212 281)(203 290 213 300)(204 289 214 299)(205 288 215 298)(206 287 216 297)(207 286 217 296)(208 285 218 295)(209 284 219 294)(210 283 220 293)(261 314 271 304)(262 313 272 303)(263 312 273 302)(264 311 274 301)(265 310 275 320)(266 309 276 319)(267 308 277 318)(268 307 278 317)(269 306 279 316)(270 305 280 315)

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

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

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

104 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P5A5B8A···8H8I8J8K8L8M···8T10A···10F20A···20X40A···40AF
order1222444444444···4558···888888···810···1020···2040···40
size11111111222210···10221···1222210···102···22···22···2

104 irreducible representations

dim111111111122222222222
type++++++-+++-
imageC1C2C2C2C2C2C4C4C4C8Q8D5C4○D4D10D10C8○D4Dic10C4×D5C8×D5C4○D20D20.3C4
kernelC8×Dic10C203C8C8×Dic5C20.8Q8C4×C40C4×Dic10C10.D4C4⋊Dic5C2×Dic10Dic10C40C4×C8C20C42C2×C8C10C8C2×C4C4C4C2
# reps112211422162222448816816

Matrix representation of C8×Dic10 in GL4(𝔽41) generated by

27000
02700
0030
0003
,
0100
403400
00911
003014
,
271400
301400
00032
00320
G:=sub<GL(4,GF(41))| [27,0,0,0,0,27,0,0,0,0,3,0,0,0,0,3],[0,40,0,0,1,34,0,0,0,0,9,30,0,0,11,14],[27,30,0,0,14,14,0,0,0,0,0,32,0,0,32,0] >;

C8×Dic10 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_{10}
% in TeX

G:=Group("C8xDic10");
// GroupNames label

G:=SmallGroup(320,305);
// by ID

G=gap.SmallGroup(320,305);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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