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G = C402C8order 320 = 26·5

2nd semidirect product of C40 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C402C8, C20.8M4(2), Dic5.11SD16, C82(C5⋊C8), C51(C82C8), (C2×C8).14F5, C10.3(C4⋊C8), C20.40(C2×C8), (C2×C40).14C4, C20⋊C8.7C2, C4.5(C4.F5), C2.1(C40⋊C4), C2.3(C20⋊C8), C10.3(C4.Q8), (C8×Dic5).21C2, (C2×Dic5).27Q8, C22.15(C4⋊F5), C2.1(C40.C4), C10.1(C8.C4), (C2×Dic5).170D4, (C4×Dic5).340C22, C4.6(C2×C5⋊C8), (C2×C52C8).19C4, (C2×C10).8(C4⋊C4), (C2×C4).117(C2×F5), (C2×C20).114(C2×C4), SmallGroup(320,219)

Series: Derived Chief Lower central Upper central

C1C20 — C402C8
C1C5C10C2×C10C2×Dic5C4×Dic5C20⋊C8 — C402C8
C5C10C20 — C402C8
C1C22C2×C4C2×C8

Generators and relations for C402C8
 G = < a,b | a40=b8=1, bab-1=a3 >

5C4
5C4
10C4
5C2×C4
5C2×C4
10C8
20C8
20C8
2Dic5
5C2×C8
5C42
10C2×C8
10C2×C8
2C52C8
4C5⋊C8
4C5⋊C8
5C4⋊C8
5C4×C8
5C4⋊C8
2C2×C5⋊C8
2C2×C5⋊C8
5C82C8

Smallest permutation representation of C402C8
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)
(1 239 104 134 200 67 254 310)(2 226 113 137 161 54 263 313)(3 213 82 140 162 41 272 316)(4 240 91 143 163 68 241 319)(5 227 100 146 164 55 250 282)(6 214 109 149 165 42 259 285)(7 201 118 152 166 69 268 288)(8 228 87 155 167 56 277 291)(9 215 96 158 168 43 246 294)(10 202 105 121 169 70 255 297)(11 229 114 124 170 57 264 300)(12 216 83 127 171 44 273 303)(13 203 92 130 172 71 242 306)(14 230 101 133 173 58 251 309)(15 217 110 136 174 45 260 312)(16 204 119 139 175 72 269 315)(17 231 88 142 176 59 278 318)(18 218 97 145 177 46 247 281)(19 205 106 148 178 73 256 284)(20 232 115 151 179 60 265 287)(21 219 84 154 180 47 274 290)(22 206 93 157 181 74 243 293)(23 233 102 160 182 61 252 296)(24 220 111 123 183 48 261 299)(25 207 120 126 184 75 270 302)(26 234 89 129 185 62 279 305)(27 221 98 132 186 49 248 308)(28 208 107 135 187 76 257 311)(29 235 116 138 188 63 266 314)(30 222 85 141 189 50 275 317)(31 209 94 144 190 77 244 320)(32 236 103 147 191 64 253 283)(33 223 112 150 192 51 262 286)(34 210 81 153 193 78 271 289)(35 237 90 156 194 65 280 292)(36 224 99 159 195 52 249 295)(37 211 108 122 196 79 258 298)(38 238 117 125 197 66 267 301)(39 225 86 128 198 53 276 304)(40 212 95 131 199 80 245 307)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H 5 8A8B8C8D8E8F8G8H8I···8P10A10B10C20A20B20C20D40A···40H
order1222444444445888888888···81010102020202040···40
size11112255551010422221010101020···2044444444···4

44 irreducible representations

dim111111222224444444
type++++-+-+
imageC1C2C2C4C4C8D4Q8SD16M4(2)C8.C4F5C5⋊C8C2×F5C4.F5C4⋊F5C40⋊C4C40.C4
kernelC402C8C8×Dic5C20⋊C8C2×C52C8C2×C40C40C2×Dic5C2×Dic5Dic5C20C10C2×C8C8C2×C4C4C22C2C2
# reps112228114241212244

Matrix representation of C402C8 in GL6(𝔽41)

28310000
17130000
0005385
003636033
008338
003303636
,
1620000
11250000
001918239
002521231
00221204
0040242022

G:=sub<GL(6,GF(41))| [28,17,0,0,0,0,31,13,0,0,0,0,0,0,0,36,8,33,0,0,5,36,3,0,0,0,38,0,3,36,0,0,5,33,8,36],[16,11,0,0,0,0,2,25,0,0,0,0,0,0,19,25,2,40,0,0,18,21,21,24,0,0,2,23,20,20,0,0,39,1,4,22] >;

C402C8 in GAP, Magma, Sage, TeX

C_{40}\rtimes_2C_8
% in TeX

G:=Group("C40:2C8");
// GroupNames label

G:=SmallGroup(320,219);
// by ID

G=gap.SmallGroup(320,219);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,6278,3156]);
// Polycyclic

G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C402C8 in TeX

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