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G = C413C8order 328 = 23·41

The semidirect product of C41 and C8 acting via C8/C4=C2

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

Aliases: C413C8, C82.2C4, C4.2D41, C2.Dic41, C164.2C2, SmallGroup(328,1)

Series: Derived Chief Lower central Upper central

C1C41 — C413C8
C1C41C82C164 — C413C8
C41 — C413C8
C1C4

Generators and relations for C413C8
 G = < a,b | a41=b8=1, bab-1=a-1 >

41C8

Smallest permutation representation of C413C8
Regular action on 328 points
Generators in S328
(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)(288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328)
(1 288 127 206 50 247 83 165)(2 328 128 246 51 287 84 205)(3 327 129 245 52 286 85 204)(4 326 130 244 53 285 86 203)(5 325 131 243 54 284 87 202)(6 324 132 242 55 283 88 201)(7 323 133 241 56 282 89 200)(8 322 134 240 57 281 90 199)(9 321 135 239 58 280 91 198)(10 320 136 238 59 279 92 197)(11 319 137 237 60 278 93 196)(12 318 138 236 61 277 94 195)(13 317 139 235 62 276 95 194)(14 316 140 234 63 275 96 193)(15 315 141 233 64 274 97 192)(16 314 142 232 65 273 98 191)(17 313 143 231 66 272 99 190)(18 312 144 230 67 271 100 189)(19 311 145 229 68 270 101 188)(20 310 146 228 69 269 102 187)(21 309 147 227 70 268 103 186)(22 308 148 226 71 267 104 185)(23 307 149 225 72 266 105 184)(24 306 150 224 73 265 106 183)(25 305 151 223 74 264 107 182)(26 304 152 222 75 263 108 181)(27 303 153 221 76 262 109 180)(28 302 154 220 77 261 110 179)(29 301 155 219 78 260 111 178)(30 300 156 218 79 259 112 177)(31 299 157 217 80 258 113 176)(32 298 158 216 81 257 114 175)(33 297 159 215 82 256 115 174)(34 296 160 214 42 255 116 173)(35 295 161 213 43 254 117 172)(36 294 162 212 44 253 118 171)(37 293 163 211 45 252 119 170)(38 292 164 210 46 251 120 169)(39 291 124 209 47 250 121 168)(40 290 125 208 48 249 122 167)(41 289 126 207 49 248 123 166)

G:=sub<Sym(328)| (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)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,288,127,206,50,247,83,165)(2,328,128,246,51,287,84,205)(3,327,129,245,52,286,85,204)(4,326,130,244,53,285,86,203)(5,325,131,243,54,284,87,202)(6,324,132,242,55,283,88,201)(7,323,133,241,56,282,89,200)(8,322,134,240,57,281,90,199)(9,321,135,239,58,280,91,198)(10,320,136,238,59,279,92,197)(11,319,137,237,60,278,93,196)(12,318,138,236,61,277,94,195)(13,317,139,235,62,276,95,194)(14,316,140,234,63,275,96,193)(15,315,141,233,64,274,97,192)(16,314,142,232,65,273,98,191)(17,313,143,231,66,272,99,190)(18,312,144,230,67,271,100,189)(19,311,145,229,68,270,101,188)(20,310,146,228,69,269,102,187)(21,309,147,227,70,268,103,186)(22,308,148,226,71,267,104,185)(23,307,149,225,72,266,105,184)(24,306,150,224,73,265,106,183)(25,305,151,223,74,264,107,182)(26,304,152,222,75,263,108,181)(27,303,153,221,76,262,109,180)(28,302,154,220,77,261,110,179)(29,301,155,219,78,260,111,178)(30,300,156,218,79,259,112,177)(31,299,157,217,80,258,113,176)(32,298,158,216,81,257,114,175)(33,297,159,215,82,256,115,174)(34,296,160,214,42,255,116,173)(35,295,161,213,43,254,117,172)(36,294,162,212,44,253,118,171)(37,293,163,211,45,252,119,170)(38,292,164,210,46,251,120,169)(39,291,124,209,47,250,121,168)(40,290,125,208,48,249,122,167)(41,289,126,207,49,248,123,166)>;

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)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,288,127,206,50,247,83,165)(2,328,128,246,51,287,84,205)(3,327,129,245,52,286,85,204)(4,326,130,244,53,285,86,203)(5,325,131,243,54,284,87,202)(6,324,132,242,55,283,88,201)(7,323,133,241,56,282,89,200)(8,322,134,240,57,281,90,199)(9,321,135,239,58,280,91,198)(10,320,136,238,59,279,92,197)(11,319,137,237,60,278,93,196)(12,318,138,236,61,277,94,195)(13,317,139,235,62,276,95,194)(14,316,140,234,63,275,96,193)(15,315,141,233,64,274,97,192)(16,314,142,232,65,273,98,191)(17,313,143,231,66,272,99,190)(18,312,144,230,67,271,100,189)(19,311,145,229,68,270,101,188)(20,310,146,228,69,269,102,187)(21,309,147,227,70,268,103,186)(22,308,148,226,71,267,104,185)(23,307,149,225,72,266,105,184)(24,306,150,224,73,265,106,183)(25,305,151,223,74,264,107,182)(26,304,152,222,75,263,108,181)(27,303,153,221,76,262,109,180)(28,302,154,220,77,261,110,179)(29,301,155,219,78,260,111,178)(30,300,156,218,79,259,112,177)(31,299,157,217,80,258,113,176)(32,298,158,216,81,257,114,175)(33,297,159,215,82,256,115,174)(34,296,160,214,42,255,116,173)(35,295,161,213,43,254,117,172)(36,294,162,212,44,253,118,171)(37,293,163,211,45,252,119,170)(38,292,164,210,46,251,120,169)(39,291,124,209,47,250,121,168)(40,290,125,208,48,249,122,167)(41,289,126,207,49,248,123,166) );

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),(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328)], [(1,288,127,206,50,247,83,165),(2,328,128,246,51,287,84,205),(3,327,129,245,52,286,85,204),(4,326,130,244,53,285,86,203),(5,325,131,243,54,284,87,202),(6,324,132,242,55,283,88,201),(7,323,133,241,56,282,89,200),(8,322,134,240,57,281,90,199),(9,321,135,239,58,280,91,198),(10,320,136,238,59,279,92,197),(11,319,137,237,60,278,93,196),(12,318,138,236,61,277,94,195),(13,317,139,235,62,276,95,194),(14,316,140,234,63,275,96,193),(15,315,141,233,64,274,97,192),(16,314,142,232,65,273,98,191),(17,313,143,231,66,272,99,190),(18,312,144,230,67,271,100,189),(19,311,145,229,68,270,101,188),(20,310,146,228,69,269,102,187),(21,309,147,227,70,268,103,186),(22,308,148,226,71,267,104,185),(23,307,149,225,72,266,105,184),(24,306,150,224,73,265,106,183),(25,305,151,223,74,264,107,182),(26,304,152,222,75,263,108,181),(27,303,153,221,76,262,109,180),(28,302,154,220,77,261,110,179),(29,301,155,219,78,260,111,178),(30,300,156,218,79,259,112,177),(31,299,157,217,80,258,113,176),(32,298,158,216,81,257,114,175),(33,297,159,215,82,256,115,174),(34,296,160,214,42,255,116,173),(35,295,161,213,43,254,117,172),(36,294,162,212,44,253,118,171),(37,293,163,211,45,252,119,170),(38,292,164,210,46,251,120,169),(39,291,124,209,47,250,121,168),(40,290,125,208,48,249,122,167),(41,289,126,207,49,248,123,166)])

88 conjugacy classes

class 1  2 4A4B8A8B8C8D41A···41T82A···82T164A···164AN
order1244888841···4182···82164···164
size1111414141412···22···22···2

88 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D41Dic41C413C8
kernelC413C8C164C82C41C4C2C1
# reps1124202040

Matrix representation of C413C8 in GL2(𝔽2297) generated by

01
2296316
,
1581449
1596716
G:=sub<GL(2,GF(2297))| [0,2296,1,316],[1581,1596,449,716] >;

C413C8 in GAP, Magma, Sage, TeX

C_{41}\rtimes_3C_8
% in TeX

G:=Group("C41:3C8");
// GroupNames label

G:=SmallGroup(328,1);
// by ID

G=gap.SmallGroup(328,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,8,21,5123]);
// Polycyclic

G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C413C8 in TeX

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