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G = C21⋊C16order 336 = 24·3·7

1st semidirect product of C21 and C16 acting via C16/C8=C2

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

Aliases: C211C16, C42.1C8, C84.2C4, C56.2S3, C24.3D7, C8.2D21, C168.3C2, C28.2Dic3, C12.2Dic7, C4.2Dic21, C3⋊(C7⋊C16), C7⋊(C3⋊C16), C6.(C7⋊C8), C14.(C3⋊C8), C2.(C21⋊C8), SmallGroup(336,5)

Series: Derived Chief Lower central Upper central

C1C21 — C21⋊C16
C1C7C21C42C84C168 — C21⋊C16
C21 — C21⋊C16
C1C8

Generators and relations for C21⋊C16
 G = < a,b | a21=b16=1, bab-1=a-1 >

21C16
7C3⋊C16
3C7⋊C16

Smallest permutation representation of C21⋊C16
Regular action on 336 points
Generators in S336
(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)
(1 325 157 249 83 289 117 209 38 306 143 228 54 264 86 188)(2 324 158 248 84 288 118 208 39 305 144 227 55 263 87 187)(3 323 159 247 64 287 119 207 40 304 145 226 56 262 88 186)(4 322 160 246 65 286 120 206 41 303 146 225 57 261 89 185)(5 321 161 245 66 285 121 205 42 302 147 224 58 260 90 184)(6 320 162 244 67 284 122 204 22 301 127 223 59 259 91 183)(7 319 163 243 68 283 123 203 23 300 128 222 60 258 92 182)(8 318 164 242 69 282 124 202 24 299 129 221 61 257 93 181)(9 317 165 241 70 281 125 201 25 298 130 220 62 256 94 180)(10 316 166 240 71 280 126 200 26 297 131 219 63 255 95 179)(11 336 167 239 72 279 106 199 27 296 132 218 43 254 96 178)(12 335 168 238 73 278 107 198 28 295 133 217 44 253 97 177)(13 334 148 237 74 277 108 197 29 315 134 216 45 273 98 176)(14 333 149 236 75 276 109 196 30 314 135 215 46 272 99 175)(15 332 150 235 76 275 110 195 31 313 136 214 47 271 100 174)(16 331 151 234 77 274 111 194 32 312 137 213 48 270 101 173)(17 330 152 233 78 294 112 193 33 311 138 212 49 269 102 172)(18 329 153 232 79 293 113 192 34 310 139 211 50 268 103 171)(19 328 154 252 80 292 114 191 35 309 140 231 51 267 104 170)(20 327 155 251 81 291 115 190 36 308 141 230 52 266 105 169)(21 326 156 250 82 290 116 210 37 307 142 229 53 265 85 189)

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

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

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

96 conjugacy classes

class 1  2  3 4A4B 6 7A7B7C8A8B8C8D12A12B14A14B14C16A···16H21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order1234467778888121214141416···1621···212424242428···2842···4256···5684···84168···168
size11211222211112222221···212···222222···22···22···22···22···2

96 irreducible representations

dim11111222222222222
type+++-+-+-
imageC1C2C4C8C16S3Dic3D7C3⋊C8Dic7D21C3⋊C16C7⋊C8Dic21C7⋊C16C21⋊C8C21⋊C16
kernelC21⋊C16C168C84C42C21C56C28C24C14C12C8C7C6C4C3C2C1
# reps11248113236466121224

Matrix representation of C21⋊C16 in GL2(𝔽41) generated by

3816
2219
,
03
10
G:=sub<GL(2,GF(41))| [38,22,16,19],[0,1,3,0] >;

C21⋊C16 in GAP, Magma, Sage, TeX

C_{21}\rtimes C_{16}
% in TeX

G:=Group("C21:C16");
// GroupNames label

G:=SmallGroup(336,5);
// by ID

G=gap.SmallGroup(336,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,12,31,50,964,10373]);
// Polycyclic

G:=Group<a,b|a^21=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C21⋊C16 in TeX

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