Copied to
clipboard

G = C5⋊C64order 320 = 26·5

The semidirect product of C5 and C64 acting via C64/C16=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5⋊C64, C10.C32, C40.2C8, C80.3C4, C16.4F5, C20.2C16, C2.(C5⋊C32), C8.4(C5⋊C8), C4.2(C5⋊C16), C52C32.2C2, SmallGroup(320,3)

Series: Derived Chief Lower central Upper central

C1C5 — C5⋊C64
C1C5C10C20C40C80C52C32 — C5⋊C64
C5 — C5⋊C64
C1C16

Generators and relations for C5⋊C64
 G = < a,b | a5=b64=1, bab-1=a3 >

5C32
5C64

Smallest permutation representation of C5⋊C64
Regular action on 320 points
Generators in S320
(1 249 78 260 131)(2 261 250 132 79)(3 133 262 80 251)(4 81 134 252 263)(5 253 82 264 135)(6 265 254 136 83)(7 137 266 84 255)(8 85 138 256 267)(9 193 86 268 139)(10 269 194 140 87)(11 141 270 88 195)(12 89 142 196 271)(13 197 90 272 143)(14 273 198 144 91)(15 145 274 92 199)(16 93 146 200 275)(17 201 94 276 147)(18 277 202 148 95)(19 149 278 96 203)(20 97 150 204 279)(21 205 98 280 151)(22 281 206 152 99)(23 153 282 100 207)(24 101 154 208 283)(25 209 102 284 155)(26 285 210 156 103)(27 157 286 104 211)(28 105 158 212 287)(29 213 106 288 159)(30 289 214 160 107)(31 161 290 108 215)(32 109 162 216 291)(33 217 110 292 163)(34 293 218 164 111)(35 165 294 112 219)(36 113 166 220 295)(37 221 114 296 167)(38 297 222 168 115)(39 169 298 116 223)(40 117 170 224 299)(41 225 118 300 171)(42 301 226 172 119)(43 173 302 120 227)(44 121 174 228 303)(45 229 122 304 175)(46 305 230 176 123)(47 177 306 124 231)(48 125 178 232 307)(49 233 126 308 179)(50 309 234 180 127)(51 181 310 128 235)(52 65 182 236 311)(53 237 66 312 183)(54 313 238 184 67)(55 185 314 68 239)(56 69 186 240 315)(57 241 70 316 187)(58 317 242 188 71)(59 189 318 72 243)(60 73 190 244 319)(61 245 74 320 191)(62 257 246 192 75)(63 129 258 76 247)(64 77 130 248 259)
(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)

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

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

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

80 conjugacy classes

class 1  2 4A4B 5 8A8B8C8D 10 16A···16H20A20B32A···32P40A40B40C40D64A···64AF80A···80H
order1244588881016···16202032···324040404064···6480···80
size11114111141···1445···544445···54···4

80 irreducible representations

dim111111144444
type+++-
imageC1C2C4C8C16C32C64F5C5⋊C8C5⋊C16C5⋊C32C5⋊C64
kernelC5⋊C64C52C32C80C40C20C10C5C16C8C4C2C1
# reps11248163211248

Matrix representation of C5⋊C64 in GL4(𝔽641) generated by

640100
640010
640001
640000
,
636292365504
360155491499
486150142223
1375155349
G:=sub<GL(4,GF(641))| [640,640,640,640,1,0,0,0,0,1,0,0,0,0,1,0],[636,360,486,137,292,155,150,515,365,491,142,5,504,499,223,349] >;

C5⋊C64 in GAP, Magma, Sage, TeX

C_5\rtimes C_{64}
% in TeX

G:=Group("C5:C64");
// GroupNames label

G:=SmallGroup(320,3);
// by ID

G=gap.SmallGroup(320,3);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,14,36,58,80,102,6278,6285]);
// Polycyclic

G:=Group<a,b|a^5=b^64=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C5⋊C64 in TeX

׿
×
𝔽