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G = C52C64order 320 = 26·5

The semidirect product of C5 and C64 acting via C64/C32=C2

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

Aliases: C52C64, C80.7C4, C40.8C8, C32.2D5, C20.5C16, C10.2C32, C160.3C2, C16.3Dic5, C2.(C52C32), C8.4(C52C8), C4.2(C52C16), SmallGroup(320,1)

Series: Derived Chief Lower central Upper central

C1C5 — C52C64
C1C5C10C20C40C80C160 — C52C64
C5 — C52C64
C1C32

Generators and relations for C52C64
 G = < a,b | a5=b64=1, bab-1=a-1 >

5C64

Smallest permutation representation of C52C64
Regular action on 320 points
Generators in S320
(1 262 121 151 246)(2 247 152 122 263)(3 264 123 153 248)(4 249 154 124 265)(5 266 125 155 250)(6 251 156 126 267)(7 268 127 157 252)(8 253 158 128 269)(9 270 65 159 254)(10 255 160 66 271)(11 272 67 161 256)(12 193 162 68 273)(13 274 69 163 194)(14 195 164 70 275)(15 276 71 165 196)(16 197 166 72 277)(17 278 73 167 198)(18 199 168 74 279)(19 280 75 169 200)(20 201 170 76 281)(21 282 77 171 202)(22 203 172 78 283)(23 284 79 173 204)(24 205 174 80 285)(25 286 81 175 206)(26 207 176 82 287)(27 288 83 177 208)(28 209 178 84 289)(29 290 85 179 210)(30 211 180 86 291)(31 292 87 181 212)(32 213 182 88 293)(33 294 89 183 214)(34 215 184 90 295)(35 296 91 185 216)(36 217 186 92 297)(37 298 93 187 218)(38 219 188 94 299)(39 300 95 189 220)(40 221 190 96 301)(41 302 97 191 222)(42 223 192 98 303)(43 304 99 129 224)(44 225 130 100 305)(45 306 101 131 226)(46 227 132 102 307)(47 308 103 133 228)(48 229 134 104 309)(49 310 105 135 230)(50 231 136 106 311)(51 312 107 137 232)(52 233 138 108 313)(53 314 109 139 234)(54 235 140 110 315)(55 316 111 141 236)(56 237 142 112 317)(57 318 113 143 238)(58 239 144 114 319)(59 320 115 145 240)(60 241 146 116 257)(61 258 117 147 242)(62 243 148 118 259)(63 260 119 149 244)(64 245 150 120 261)
(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)| (1,262,121,151,246)(2,247,152,122,263)(3,264,123,153,248)(4,249,154,124,265)(5,266,125,155,250)(6,251,156,126,267)(7,268,127,157,252)(8,253,158,128,269)(9,270,65,159,254)(10,255,160,66,271)(11,272,67,161,256)(12,193,162,68,273)(13,274,69,163,194)(14,195,164,70,275)(15,276,71,165,196)(16,197,166,72,277)(17,278,73,167,198)(18,199,168,74,279)(19,280,75,169,200)(20,201,170,76,281)(21,282,77,171,202)(22,203,172,78,283)(23,284,79,173,204)(24,205,174,80,285)(25,286,81,175,206)(26,207,176,82,287)(27,288,83,177,208)(28,209,178,84,289)(29,290,85,179,210)(30,211,180,86,291)(31,292,87,181,212)(32,213,182,88,293)(33,294,89,183,214)(34,215,184,90,295)(35,296,91,185,216)(36,217,186,92,297)(37,298,93,187,218)(38,219,188,94,299)(39,300,95,189,220)(40,221,190,96,301)(41,302,97,191,222)(42,223,192,98,303)(43,304,99,129,224)(44,225,130,100,305)(45,306,101,131,226)(46,227,132,102,307)(47,308,103,133,228)(48,229,134,104,309)(49,310,105,135,230)(50,231,136,106,311)(51,312,107,137,232)(52,233,138,108,313)(53,314,109,139,234)(54,235,140,110,315)(55,316,111,141,236)(56,237,142,112,317)(57,318,113,143,238)(58,239,144,114,319)(59,320,115,145,240)(60,241,146,116,257)(61,258,117,147,242)(62,243,148,118,259)(63,260,119,149,244)(64,245,150,120,261), (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( (1,262,121,151,246)(2,247,152,122,263)(3,264,123,153,248)(4,249,154,124,265)(5,266,125,155,250)(6,251,156,126,267)(7,268,127,157,252)(8,253,158,128,269)(9,270,65,159,254)(10,255,160,66,271)(11,272,67,161,256)(12,193,162,68,273)(13,274,69,163,194)(14,195,164,70,275)(15,276,71,165,196)(16,197,166,72,277)(17,278,73,167,198)(18,199,168,74,279)(19,280,75,169,200)(20,201,170,76,281)(21,282,77,171,202)(22,203,172,78,283)(23,284,79,173,204)(24,205,174,80,285)(25,286,81,175,206)(26,207,176,82,287)(27,288,83,177,208)(28,209,178,84,289)(29,290,85,179,210)(30,211,180,86,291)(31,292,87,181,212)(32,213,182,88,293)(33,294,89,183,214)(34,215,184,90,295)(35,296,91,185,216)(36,217,186,92,297)(37,298,93,187,218)(38,219,188,94,299)(39,300,95,189,220)(40,221,190,96,301)(41,302,97,191,222)(42,223,192,98,303)(43,304,99,129,224)(44,225,130,100,305)(45,306,101,131,226)(46,227,132,102,307)(47,308,103,133,228)(48,229,134,104,309)(49,310,105,135,230)(50,231,136,106,311)(51,312,107,137,232)(52,233,138,108,313)(53,314,109,139,234)(54,235,140,110,315)(55,316,111,141,236)(56,237,142,112,317)(57,318,113,143,238)(58,239,144,114,319)(59,320,115,145,240)(60,241,146,116,257)(61,258,117,147,242)(62,243,148,118,259)(63,260,119,149,244)(64,245,150,120,261), (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([(1,262,121,151,246),(2,247,152,122,263),(3,264,123,153,248),(4,249,154,124,265),(5,266,125,155,250),(6,251,156,126,267),(7,268,127,157,252),(8,253,158,128,269),(9,270,65,159,254),(10,255,160,66,271),(11,272,67,161,256),(12,193,162,68,273),(13,274,69,163,194),(14,195,164,70,275),(15,276,71,165,196),(16,197,166,72,277),(17,278,73,167,198),(18,199,168,74,279),(19,280,75,169,200),(20,201,170,76,281),(21,282,77,171,202),(22,203,172,78,283),(23,284,79,173,204),(24,205,174,80,285),(25,286,81,175,206),(26,207,176,82,287),(27,288,83,177,208),(28,209,178,84,289),(29,290,85,179,210),(30,211,180,86,291),(31,292,87,181,212),(32,213,182,88,293),(33,294,89,183,214),(34,215,184,90,295),(35,296,91,185,216),(36,217,186,92,297),(37,298,93,187,218),(38,219,188,94,299),(39,300,95,189,220),(40,221,190,96,301),(41,302,97,191,222),(42,223,192,98,303),(43,304,99,129,224),(44,225,130,100,305),(45,306,101,131,226),(46,227,132,102,307),(47,308,103,133,228),(48,229,134,104,309),(49,310,105,135,230),(50,231,136,106,311),(51,312,107,137,232),(52,233,138,108,313),(53,314,109,139,234),(54,235,140,110,315),(55,316,111,141,236),(56,237,142,112,317),(57,318,113,143,238),(58,239,144,114,319),(59,320,115,145,240),(60,241,146,116,257),(61,258,117,147,242),(62,243,148,118,259),(63,260,119,149,244),(64,245,150,120,261)], [(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)])

128 conjugacy classes

class 1  2 4A4B5A5B8A8B8C8D10A10B16A···16H20A20B20C20D32A···32P40A···40H64A···64AF80A···80P160A···160AF
order1244558888101016···162020202032···3240···4064···6480···80160···160
size1111221111221···122221···12···25···52···22···2

128 irreducible representations

dim1111111222222
type+++-
imageC1C2C4C8C16C32C64D5Dic5C52C8C52C16C52C32C52C64
kernelC52C64C160C80C40C20C10C5C32C16C8C4C2C1
# reps11248163222481632

Matrix representation of C52C64 in GL2(𝔽641) generated by

01
640362
,
152591
489489
G:=sub<GL(2,GF(641))| [0,640,1,362],[152,489,591,489] >;

C52C64 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_{64}
% in TeX

G:=Group("C5:2C64");
// GroupNames label

G:=SmallGroup(320,1);
// by ID

G=gap.SmallGroup(320,1);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,14,36,58,80,102,12550]);
// Polycyclic

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

Export

Subgroup lattice of C52C64 in TeX

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