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G = C33⋊C8order 264 = 23·3·11

1st semidirect product of C33 and C8 acting via C8/C4=C2

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

Aliases: C331C8, C66.1C4, C44.2S3, C4.2D33, C22.Dic3, C2.Dic33, C6.Dic11, C132.2C2, C12.2D11, C11⋊(C3⋊C8), C3⋊(C11⋊C8), SmallGroup(264,3)

Series: Derived Chief Lower central Upper central

C1C33 — C33⋊C8
C1C11C33C66C132 — C33⋊C8
C33 — C33⋊C8
C1C4

Generators and relations for C33⋊C8
 G = < a,b | a33=b8=1, bab-1=a-1 >

33C8
11C3⋊C8
3C11⋊C8

Smallest permutation representation of C33⋊C8
Regular action on 264 points
Generators in S264
(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)
(1 259 100 188 42 222 83 164)(2 258 101 187 43 221 84 163)(3 257 102 186 44 220 85 162)(4 256 103 185 45 219 86 161)(5 255 104 184 46 218 87 160)(6 254 105 183 47 217 88 159)(7 253 106 182 48 216 89 158)(8 252 107 181 49 215 90 157)(9 251 108 180 50 214 91 156)(10 250 109 179 51 213 92 155)(11 249 110 178 52 212 93 154)(12 248 111 177 53 211 94 153)(13 247 112 176 54 210 95 152)(14 246 113 175 55 209 96 151)(15 245 114 174 56 208 97 150)(16 244 115 173 57 207 98 149)(17 243 116 172 58 206 99 148)(18 242 117 171 59 205 67 147)(19 241 118 170 60 204 68 146)(20 240 119 169 61 203 69 145)(21 239 120 168 62 202 70 144)(22 238 121 167 63 201 71 143)(23 237 122 166 64 200 72 142)(24 236 123 198 65 199 73 141)(25 235 124 197 66 231 74 140)(26 234 125 196 34 230 75 139)(27 233 126 195 35 229 76 138)(28 232 127 194 36 228 77 137)(29 264 128 193 37 227 78 136)(30 263 129 192 38 226 79 135)(31 262 130 191 39 225 80 134)(32 261 131 190 40 224 81 133)(33 260 132 189 41 223 82 165)

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

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

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

72 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D11A···11E12A12B22A···22E33A···33J44A···44J66A···66J132A···132T
order123446888811···11121222···2233···3344···4466···66132···132
size112112333333332···2222···22···22···22···22···2

72 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3D11C3⋊C8Dic11D33C11⋊C8Dic33C33⋊C8
kernelC33⋊C8C132C66C33C44C22C12C11C6C4C3C2C1
# reps11241152510101020

Matrix representation of C33⋊C8 in GL2(𝔽1321) generated by

8471079
877863
,
315300
5861006
G:=sub<GL(2,GF(1321))| [847,877,1079,863],[315,586,300,1006] >;

C33⋊C8 in GAP, Magma, Sage, TeX

C_{33}\rtimes C_8
% in TeX

G:=Group("C33:C8");
// GroupNames label

G:=SmallGroup(264,3);
// by ID

G=gap.SmallGroup(264,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,10,26,323,6004]);
// Polycyclic

G:=Group<a,b|a^33=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C33⋊C8 in TeX

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