Copied to
clipboard

G = C43⋊C8order 344 = 23·43

The semidirect product of C43 and C8 acting via C8/C4=C2

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

Aliases: C43⋊C8, C86.C4, C4.2D43, C2.Dic43, C172.2C2, SmallGroup(344,1)

Series: Derived Chief Lower central Upper central

C1C43 — C43⋊C8
C1C43C86C172 — C43⋊C8
C43 — C43⋊C8
C1C4

Generators and relations for C43⋊C8
 G = < a,b | a43=b8=1, bab-1=a-1 >

43C8

Smallest permutation representation of C43⋊C8
Regular action on 344 points
Generators in S344
(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)(302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344)
(1 302 160 216 68 259 101 173)(2 344 161 258 69 301 102 215)(3 343 162 257 70 300 103 214)(4 342 163 256 71 299 104 213)(5 341 164 255 72 298 105 212)(6 340 165 254 73 297 106 211)(7 339 166 253 74 296 107 210)(8 338 167 252 75 295 108 209)(9 337 168 251 76 294 109 208)(10 336 169 250 77 293 110 207)(11 335 170 249 78 292 111 206)(12 334 171 248 79 291 112 205)(13 333 172 247 80 290 113 204)(14 332 130 246 81 289 114 203)(15 331 131 245 82 288 115 202)(16 330 132 244 83 287 116 201)(17 329 133 243 84 286 117 200)(18 328 134 242 85 285 118 199)(19 327 135 241 86 284 119 198)(20 326 136 240 44 283 120 197)(21 325 137 239 45 282 121 196)(22 324 138 238 46 281 122 195)(23 323 139 237 47 280 123 194)(24 322 140 236 48 279 124 193)(25 321 141 235 49 278 125 192)(26 320 142 234 50 277 126 191)(27 319 143 233 51 276 127 190)(28 318 144 232 52 275 128 189)(29 317 145 231 53 274 129 188)(30 316 146 230 54 273 87 187)(31 315 147 229 55 272 88 186)(32 314 148 228 56 271 89 185)(33 313 149 227 57 270 90 184)(34 312 150 226 58 269 91 183)(35 311 151 225 59 268 92 182)(36 310 152 224 60 267 93 181)(37 309 153 223 61 266 94 180)(38 308 154 222 62 265 95 179)(39 307 155 221 63 264 96 178)(40 306 156 220 64 263 97 177)(41 305 157 219 65 262 98 176)(42 304 158 218 66 261 99 175)(43 303 159 217 67 260 100 174)

G:=sub<Sym(344)| (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)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,302,160,216,68,259,101,173)(2,344,161,258,69,301,102,215)(3,343,162,257,70,300,103,214)(4,342,163,256,71,299,104,213)(5,341,164,255,72,298,105,212)(6,340,165,254,73,297,106,211)(7,339,166,253,74,296,107,210)(8,338,167,252,75,295,108,209)(9,337,168,251,76,294,109,208)(10,336,169,250,77,293,110,207)(11,335,170,249,78,292,111,206)(12,334,171,248,79,291,112,205)(13,333,172,247,80,290,113,204)(14,332,130,246,81,289,114,203)(15,331,131,245,82,288,115,202)(16,330,132,244,83,287,116,201)(17,329,133,243,84,286,117,200)(18,328,134,242,85,285,118,199)(19,327,135,241,86,284,119,198)(20,326,136,240,44,283,120,197)(21,325,137,239,45,282,121,196)(22,324,138,238,46,281,122,195)(23,323,139,237,47,280,123,194)(24,322,140,236,48,279,124,193)(25,321,141,235,49,278,125,192)(26,320,142,234,50,277,126,191)(27,319,143,233,51,276,127,190)(28,318,144,232,52,275,128,189)(29,317,145,231,53,274,129,188)(30,316,146,230,54,273,87,187)(31,315,147,229,55,272,88,186)(32,314,148,228,56,271,89,185)(33,313,149,227,57,270,90,184)(34,312,150,226,58,269,91,183)(35,311,151,225,59,268,92,182)(36,310,152,224,60,267,93,181)(37,309,153,223,61,266,94,180)(38,308,154,222,62,265,95,179)(39,307,155,221,63,264,96,178)(40,306,156,220,64,263,97,177)(41,305,157,219,65,262,98,176)(42,304,158,218,66,261,99,175)(43,303,159,217,67,260,100,174)>;

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)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,302,160,216,68,259,101,173)(2,344,161,258,69,301,102,215)(3,343,162,257,70,300,103,214)(4,342,163,256,71,299,104,213)(5,341,164,255,72,298,105,212)(6,340,165,254,73,297,106,211)(7,339,166,253,74,296,107,210)(8,338,167,252,75,295,108,209)(9,337,168,251,76,294,109,208)(10,336,169,250,77,293,110,207)(11,335,170,249,78,292,111,206)(12,334,171,248,79,291,112,205)(13,333,172,247,80,290,113,204)(14,332,130,246,81,289,114,203)(15,331,131,245,82,288,115,202)(16,330,132,244,83,287,116,201)(17,329,133,243,84,286,117,200)(18,328,134,242,85,285,118,199)(19,327,135,241,86,284,119,198)(20,326,136,240,44,283,120,197)(21,325,137,239,45,282,121,196)(22,324,138,238,46,281,122,195)(23,323,139,237,47,280,123,194)(24,322,140,236,48,279,124,193)(25,321,141,235,49,278,125,192)(26,320,142,234,50,277,126,191)(27,319,143,233,51,276,127,190)(28,318,144,232,52,275,128,189)(29,317,145,231,53,274,129,188)(30,316,146,230,54,273,87,187)(31,315,147,229,55,272,88,186)(32,314,148,228,56,271,89,185)(33,313,149,227,57,270,90,184)(34,312,150,226,58,269,91,183)(35,311,151,225,59,268,92,182)(36,310,152,224,60,267,93,181)(37,309,153,223,61,266,94,180)(38,308,154,222,62,265,95,179)(39,307,155,221,63,264,96,178)(40,306,156,220,64,263,97,177)(41,305,157,219,65,262,98,176)(42,304,158,218,66,261,99,175)(43,303,159,217,67,260,100,174) );

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),(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344)], [(1,302,160,216,68,259,101,173),(2,344,161,258,69,301,102,215),(3,343,162,257,70,300,103,214),(4,342,163,256,71,299,104,213),(5,341,164,255,72,298,105,212),(6,340,165,254,73,297,106,211),(7,339,166,253,74,296,107,210),(8,338,167,252,75,295,108,209),(9,337,168,251,76,294,109,208),(10,336,169,250,77,293,110,207),(11,335,170,249,78,292,111,206),(12,334,171,248,79,291,112,205),(13,333,172,247,80,290,113,204),(14,332,130,246,81,289,114,203),(15,331,131,245,82,288,115,202),(16,330,132,244,83,287,116,201),(17,329,133,243,84,286,117,200),(18,328,134,242,85,285,118,199),(19,327,135,241,86,284,119,198),(20,326,136,240,44,283,120,197),(21,325,137,239,45,282,121,196),(22,324,138,238,46,281,122,195),(23,323,139,237,47,280,123,194),(24,322,140,236,48,279,124,193),(25,321,141,235,49,278,125,192),(26,320,142,234,50,277,126,191),(27,319,143,233,51,276,127,190),(28,318,144,232,52,275,128,189),(29,317,145,231,53,274,129,188),(30,316,146,230,54,273,87,187),(31,315,147,229,55,272,88,186),(32,314,148,228,56,271,89,185),(33,313,149,227,57,270,90,184),(34,312,150,226,58,269,91,183),(35,311,151,225,59,268,92,182),(36,310,152,224,60,267,93,181),(37,309,153,223,61,266,94,180),(38,308,154,222,62,265,95,179),(39,307,155,221,63,264,96,178),(40,306,156,220,64,263,97,177),(41,305,157,219,65,262,98,176),(42,304,158,218,66,261,99,175),(43,303,159,217,67,260,100,174)])

92 conjugacy classes

class 1  2 4A4B8A8B8C8D43A···43U86A···86U172A···172AP
order1244888843···4386···86172···172
size1111434343432···22···22···2

92 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D43Dic43C43⋊C8
kernelC43⋊C8C172C86C43C4C2C1
# reps1124212142

Matrix representation of C43⋊C8 in GL3(𝔽1033) generated by

100
001
01032125
,
63500
0123633
0513910
G:=sub<GL(3,GF(1033))| [1,0,0,0,0,1032,0,1,125],[635,0,0,0,123,513,0,633,910] >;

C43⋊C8 in GAP, Magma, Sage, TeX

C_{43}\rtimes C_8
% in TeX

G:=Group("C43:C8");
// GroupNames label

G:=SmallGroup(344,1);
// by ID

G=gap.SmallGroup(344,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-43,8,21,5379]);
// Polycyclic

G:=Group<a,b|a^43=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C43⋊C8 in TeX

׿
×
𝔽