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## G = C80⋊14C4order 320 = 26·5

### 2nd semidirect product of C80 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C80⋊14C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40⋊5C4 — C80⋊14C4
 Lower central C5 — C10 — C20 — C40 — C80⋊14C4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C8014C4
G = < a,b | a80=b4=1, bab-1=a39 >

Smallest permutation representation of C8014C4
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)
(1 274 186 136)(2 313 187 95)(3 272 188 134)(4 311 189 93)(5 270 190 132)(6 309 191 91)(7 268 192 130)(8 307 193 89)(9 266 194 128)(10 305 195 87)(11 264 196 126)(12 303 197 85)(13 262 198 124)(14 301 199 83)(15 260 200 122)(16 299 201 81)(17 258 202 120)(18 297 203 159)(19 256 204 118)(20 295 205 157)(21 254 206 116)(22 293 207 155)(23 252 208 114)(24 291 209 153)(25 250 210 112)(26 289 211 151)(27 248 212 110)(28 287 213 149)(29 246 214 108)(30 285 215 147)(31 244 216 106)(32 283 217 145)(33 242 218 104)(34 281 219 143)(35 320 220 102)(36 279 221 141)(37 318 222 100)(38 277 223 139)(39 316 224 98)(40 275 225 137)(41 314 226 96)(42 273 227 135)(43 312 228 94)(44 271 229 133)(45 310 230 92)(46 269 231 131)(47 308 232 90)(48 267 233 129)(49 306 234 88)(50 265 235 127)(51 304 236 86)(52 263 237 125)(53 302 238 84)(54 261 239 123)(55 300 240 82)(56 259 161 121)(57 298 162 160)(58 257 163 119)(59 296 164 158)(60 255 165 117)(61 294 166 156)(62 253 167 115)(63 292 168 154)(64 251 169 113)(65 290 170 152)(66 249 171 111)(67 288 172 150)(68 247 173 109)(69 286 174 148)(70 245 175 107)(71 284 176 146)(72 243 177 105)(73 282 178 144)(74 241 179 103)(75 280 180 142)(76 319 181 101)(77 278 182 140)(78 317 183 99)(79 276 184 138)(80 315 185 97)

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

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

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

86 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 40 40 40 40 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - + + - + - + - + - + image C1 C2 C2 C4 Q8 D4 D5 Q16 D8 Dic5 D10 SD32 Dic10 D20 Dic20 D40 C16⋊D5 kernel C80⋊14C4 C40⋊5C4 C2×C80 C80 C40 C2×C20 C2×C16 C20 C2×C10 C16 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 2 1 4 1 1 2 2 2 4 2 8 4 4 8 8 32

Matrix representation of C8014C4 in GL4(𝔽241) generated by

 56 42 0 0 172 228 0 0 0 0 79 85 0 0 156 238
,
 93 106 0 0 73 148 0 0 0 0 169 75 0 0 204 72
G:=sub<GL(4,GF(241))| [56,172,0,0,42,228,0,0,0,0,79,156,0,0,85,238],[93,73,0,0,106,148,0,0,0,0,169,204,0,0,75,72] >;

C8014C4 in GAP, Magma, Sage, TeX

C_{80}\rtimes_{14}C_4
% in TeX

G:=Group("C80:14C4");
// GroupNames label

G:=SmallGroup(320,63);
// by ID

G=gap.SmallGroup(320,63);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1571,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^39>;
// generators/relations

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