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G = C23⋊C16order 368 = 24·23

The semidirect product of C23 and C16 acting via C16/C8=C2

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

Aliases: C23⋊C16, C46.C8, C92.2C4, C8.2D23, C184.2C2, C4.2Dic23, C2.(C23⋊C8), SmallGroup(368,1)

Series: Derived Chief Lower central Upper central

C1C23 — C23⋊C16
C1C23C46C92C184 — C23⋊C16
C23 — C23⋊C16
C1C8

Generators and relations for C23⋊C16
 G = < a,b | a23=b16=1, bab-1=a-1 >

23C16

Smallest permutation representation of C23⋊C16
Regular action on 368 points
Generators in S368
(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)(323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345)(346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368)
(1 346 171 254 92 300 118 208 46 323 145 231 47 277 102 185)(2 368 172 276 70 322 119 230 24 345 146 253 48 299 103 207)(3 367 173 275 71 321 120 229 25 344 147 252 49 298 104 206)(4 366 174 274 72 320 121 228 26 343 148 251 50 297 105 205)(5 365 175 273 73 319 122 227 27 342 149 250 51 296 106 204)(6 364 176 272 74 318 123 226 28 341 150 249 52 295 107 203)(7 363 177 271 75 317 124 225 29 340 151 248 53 294 108 202)(8 362 178 270 76 316 125 224 30 339 152 247 54 293 109 201)(9 361 179 269 77 315 126 223 31 338 153 246 55 292 110 200)(10 360 180 268 78 314 127 222 32 337 154 245 56 291 111 199)(11 359 181 267 79 313 128 221 33 336 155 244 57 290 112 198)(12 358 182 266 80 312 129 220 34 335 156 243 58 289 113 197)(13 357 183 265 81 311 130 219 35 334 157 242 59 288 114 196)(14 356 184 264 82 310 131 218 36 333 158 241 60 287 115 195)(15 355 162 263 83 309 132 217 37 332 159 240 61 286 93 194)(16 354 163 262 84 308 133 216 38 331 160 239 62 285 94 193)(17 353 164 261 85 307 134 215 39 330 161 238 63 284 95 192)(18 352 165 260 86 306 135 214 40 329 139 237 64 283 96 191)(19 351 166 259 87 305 136 213 41 328 140 236 65 282 97 190)(20 350 167 258 88 304 137 212 42 327 141 235 66 281 98 189)(21 349 168 257 89 303 138 211 43 326 142 234 67 280 99 188)(22 348 169 256 90 302 116 210 44 325 143 233 68 279 100 187)(23 347 170 255 91 301 117 209 45 324 144 232 69 278 101 186)

G:=sub<Sym(368)| (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)(323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368), (1,346,171,254,92,300,118,208,46,323,145,231,47,277,102,185)(2,368,172,276,70,322,119,230,24,345,146,253,48,299,103,207)(3,367,173,275,71,321,120,229,25,344,147,252,49,298,104,206)(4,366,174,274,72,320,121,228,26,343,148,251,50,297,105,205)(5,365,175,273,73,319,122,227,27,342,149,250,51,296,106,204)(6,364,176,272,74,318,123,226,28,341,150,249,52,295,107,203)(7,363,177,271,75,317,124,225,29,340,151,248,53,294,108,202)(8,362,178,270,76,316,125,224,30,339,152,247,54,293,109,201)(9,361,179,269,77,315,126,223,31,338,153,246,55,292,110,200)(10,360,180,268,78,314,127,222,32,337,154,245,56,291,111,199)(11,359,181,267,79,313,128,221,33,336,155,244,57,290,112,198)(12,358,182,266,80,312,129,220,34,335,156,243,58,289,113,197)(13,357,183,265,81,311,130,219,35,334,157,242,59,288,114,196)(14,356,184,264,82,310,131,218,36,333,158,241,60,287,115,195)(15,355,162,263,83,309,132,217,37,332,159,240,61,286,93,194)(16,354,163,262,84,308,133,216,38,331,160,239,62,285,94,193)(17,353,164,261,85,307,134,215,39,330,161,238,63,284,95,192)(18,352,165,260,86,306,135,214,40,329,139,237,64,283,96,191)(19,351,166,259,87,305,136,213,41,328,140,236,65,282,97,190)(20,350,167,258,88,304,137,212,42,327,141,235,66,281,98,189)(21,349,168,257,89,303,138,211,43,326,142,234,67,280,99,188)(22,348,169,256,90,302,116,210,44,325,143,233,68,279,100,187)(23,347,170,255,91,301,117,209,45,324,144,232,69,278,101,186)>;

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)(323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368), (1,346,171,254,92,300,118,208,46,323,145,231,47,277,102,185)(2,368,172,276,70,322,119,230,24,345,146,253,48,299,103,207)(3,367,173,275,71,321,120,229,25,344,147,252,49,298,104,206)(4,366,174,274,72,320,121,228,26,343,148,251,50,297,105,205)(5,365,175,273,73,319,122,227,27,342,149,250,51,296,106,204)(6,364,176,272,74,318,123,226,28,341,150,249,52,295,107,203)(7,363,177,271,75,317,124,225,29,340,151,248,53,294,108,202)(8,362,178,270,76,316,125,224,30,339,152,247,54,293,109,201)(9,361,179,269,77,315,126,223,31,338,153,246,55,292,110,200)(10,360,180,268,78,314,127,222,32,337,154,245,56,291,111,199)(11,359,181,267,79,313,128,221,33,336,155,244,57,290,112,198)(12,358,182,266,80,312,129,220,34,335,156,243,58,289,113,197)(13,357,183,265,81,311,130,219,35,334,157,242,59,288,114,196)(14,356,184,264,82,310,131,218,36,333,158,241,60,287,115,195)(15,355,162,263,83,309,132,217,37,332,159,240,61,286,93,194)(16,354,163,262,84,308,133,216,38,331,160,239,62,285,94,193)(17,353,164,261,85,307,134,215,39,330,161,238,63,284,95,192)(18,352,165,260,86,306,135,214,40,329,139,237,64,283,96,191)(19,351,166,259,87,305,136,213,41,328,140,236,65,282,97,190)(20,350,167,258,88,304,137,212,42,327,141,235,66,281,98,189)(21,349,168,257,89,303,138,211,43,326,142,234,67,280,99,188)(22,348,169,256,90,302,116,210,44,325,143,233,68,279,100,187)(23,347,170,255,91,301,117,209,45,324,144,232,69,278,101,186) );

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),(323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345),(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368)], [(1,346,171,254,92,300,118,208,46,323,145,231,47,277,102,185),(2,368,172,276,70,322,119,230,24,345,146,253,48,299,103,207),(3,367,173,275,71,321,120,229,25,344,147,252,49,298,104,206),(4,366,174,274,72,320,121,228,26,343,148,251,50,297,105,205),(5,365,175,273,73,319,122,227,27,342,149,250,51,296,106,204),(6,364,176,272,74,318,123,226,28,341,150,249,52,295,107,203),(7,363,177,271,75,317,124,225,29,340,151,248,53,294,108,202),(8,362,178,270,76,316,125,224,30,339,152,247,54,293,109,201),(9,361,179,269,77,315,126,223,31,338,153,246,55,292,110,200),(10,360,180,268,78,314,127,222,32,337,154,245,56,291,111,199),(11,359,181,267,79,313,128,221,33,336,155,244,57,290,112,198),(12,358,182,266,80,312,129,220,34,335,156,243,58,289,113,197),(13,357,183,265,81,311,130,219,35,334,157,242,59,288,114,196),(14,356,184,264,82,310,131,218,36,333,158,241,60,287,115,195),(15,355,162,263,83,309,132,217,37,332,159,240,61,286,93,194),(16,354,163,262,84,308,133,216,38,331,160,239,62,285,94,193),(17,353,164,261,85,307,134,215,39,330,161,238,63,284,95,192),(18,352,165,260,86,306,135,214,40,329,139,237,64,283,96,191),(19,351,166,259,87,305,136,213,41,328,140,236,65,282,97,190),(20,350,167,258,88,304,137,212,42,327,141,235,66,281,98,189),(21,349,168,257,89,303,138,211,43,326,142,234,67,280,99,188),(22,348,169,256,90,302,116,210,44,325,143,233,68,279,100,187),(23,347,170,255,91,301,117,209,45,324,144,232,69,278,101,186)])

104 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H23A···23K46A···46K92A···92V184A···184AR
order1244888816···1623···2346···4692···92184···184
size1111111123···232···22···22···22···2

104 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D23Dic23C23⋊C8C23⋊C16
kernelC23⋊C16C184C92C46C23C8C4C2C1
# reps1124811112244

Matrix representation of C23⋊C16 in GL2(𝔽3313) generated by

01
33121146
,
5411639
20942772
G:=sub<GL(2,GF(3313))| [0,3312,1,1146],[541,2094,1639,2772] >;

C23⋊C16 in GAP, Magma, Sage, TeX

C_{23}\rtimes C_{16}
% in TeX

G:=Group("C23:C16");
// GroupNames label

G:=SmallGroup(368,1);
// by ID

G=gap.SmallGroup(368,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,10,26,42,8804]);
// Polycyclic

G:=Group<a,b|a^23=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C23⋊C16 in TeX

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