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G = Dic66order 264 = 23·3·11

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic66, C4.D33, C332Q8, C44.1S3, C6.8D22, C2.3D66, C22.8D6, C32Dic22, C112Dic6, C132.1C2, C12.1D11, C66.8C22, Dic33.1C2, SmallGroup(264,23)

Series: Derived Chief Lower central Upper central

C1C66 — Dic66
C1C11C33C66Dic33 — Dic66
C33C66 — Dic66
C1C2C4

Generators and relations for Dic66
 G = < a,b | a132=1, b2=a66, bab-1=a-1 >

33C4
33C4
33Q8
11Dic3
11Dic3
3Dic11
3Dic11
11Dic6
3Dic22

Smallest permutation representation of Dic66
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)
(1 152 67 218)(2 151 68 217)(3 150 69 216)(4 149 70 215)(5 148 71 214)(6 147 72 213)(7 146 73 212)(8 145 74 211)(9 144 75 210)(10 143 76 209)(11 142 77 208)(12 141 78 207)(13 140 79 206)(14 139 80 205)(15 138 81 204)(16 137 82 203)(17 136 83 202)(18 135 84 201)(19 134 85 200)(20 133 86 199)(21 264 87 198)(22 263 88 197)(23 262 89 196)(24 261 90 195)(25 260 91 194)(26 259 92 193)(27 258 93 192)(28 257 94 191)(29 256 95 190)(30 255 96 189)(31 254 97 188)(32 253 98 187)(33 252 99 186)(34 251 100 185)(35 250 101 184)(36 249 102 183)(37 248 103 182)(38 247 104 181)(39 246 105 180)(40 245 106 179)(41 244 107 178)(42 243 108 177)(43 242 109 176)(44 241 110 175)(45 240 111 174)(46 239 112 173)(47 238 113 172)(48 237 114 171)(49 236 115 170)(50 235 116 169)(51 234 117 168)(52 233 118 167)(53 232 119 166)(54 231 120 165)(55 230 121 164)(56 229 122 163)(57 228 123 162)(58 227 124 161)(59 226 125 160)(60 225 126 159)(61 224 127 158)(62 223 128 157)(63 222 129 156)(64 221 130 155)(65 220 131 154)(66 219 132 153)

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

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

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

69 conjugacy classes

class 1  2  3 4A4B4C 6 11A···11E12A12B22A···22E33A···33J44A···44J66A···66J132A···132T
order123444611···11121222···2233···3344···4466···66132···132
size1122666622···2222···22···22···22···22···2

69 irreducible representations

dim1112222222222
type++++-++-++-+-
imageC1C2C2S3Q8D6D11Dic6D22D33Dic22D66Dic66
kernelDic66Dic33C132C44C33C22C12C11C6C4C3C2C1
# reps12111152510101020

Matrix representation of Dic66 in GL2(𝔽397) generated by

18949
122290
,
95386
207302
G:=sub<GL(2,GF(397))| [189,122,49,290],[95,207,386,302] >;

Dic66 in GAP, Magma, Sage, TeX

{\rm Dic}_{66}
% in TeX

G:=Group("Dic66");
// GroupNames label

G:=SmallGroup(264,23);
// by ID

G=gap.SmallGroup(264,23);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,20,61,26,323,6004]);
// Polycyclic

G:=Group<a,b|a^132=1,b^2=a^66,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic66 in TeX

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