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G = Dic54order 216 = 23·33

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic54, C27⋊Q8, C4.D27, C9.Dic6, C36.1S3, C2.3D54, C6.8D18, C18.8D6, C12.1D9, C3.Dic18, C108.1C2, Dic27.C2, C54.1C22, SmallGroup(216,4)

Series: Derived Chief Lower central Upper central

C1C54 — Dic54
C1C3C9C27C54Dic27 — Dic54
C27C54 — Dic54
C1C2C4

Generators and relations for Dic54
 G = < a,b | a108=1, b2=a54, bab-1=a-1 >

27C4
27C4
27Q8
9Dic3
9Dic3
9Dic6
3Dic9
3Dic9
3Dic18

Smallest permutation representation of Dic54
Regular action on 216 points
Generators in S216
(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)
(1 216 55 162)(2 215 56 161)(3 214 57 160)(4 213 58 159)(5 212 59 158)(6 211 60 157)(7 210 61 156)(8 209 62 155)(9 208 63 154)(10 207 64 153)(11 206 65 152)(12 205 66 151)(13 204 67 150)(14 203 68 149)(15 202 69 148)(16 201 70 147)(17 200 71 146)(18 199 72 145)(19 198 73 144)(20 197 74 143)(21 196 75 142)(22 195 76 141)(23 194 77 140)(24 193 78 139)(25 192 79 138)(26 191 80 137)(27 190 81 136)(28 189 82 135)(29 188 83 134)(30 187 84 133)(31 186 85 132)(32 185 86 131)(33 184 87 130)(34 183 88 129)(35 182 89 128)(36 181 90 127)(37 180 91 126)(38 179 92 125)(39 178 93 124)(40 177 94 123)(41 176 95 122)(42 175 96 121)(43 174 97 120)(44 173 98 119)(45 172 99 118)(46 171 100 117)(47 170 101 116)(48 169 102 115)(49 168 103 114)(50 167 104 113)(51 166 105 112)(52 165 106 111)(53 164 107 110)(54 163 108 109)

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

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

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

Dic54 is a maximal subgroup of   Dic108  C216⋊C2  D4.D27  C27⋊Q16  D1085C2  D42D27  Q8×D27
Dic54 is a maximal quotient of   Dic27⋊C4  C4⋊Dic27

57 conjugacy classes

class 1  2  3 4A4B4C 6 9A9B9C12A12B18A18B18C27A···27I36A···36F54A···54I108A···108R
order1234446999121218181827···2736···3654···54108···108
size112254542222222222···22···22···22···2

57 irreducible representations

dim1112222222222
type++++-++-++-+-
imageC1C2C2S3Q8D6D9Dic6D18D27Dic18D54Dic54
kernelDic54Dic27C108C36C27C18C12C9C6C4C3C2C1
# reps12111132396918

Matrix representation of Dic54 in GL2(𝔽109) generated by

60104
565
,
4520
8464
G:=sub<GL(2,GF(109))| [60,5,104,65],[45,84,20,64] >;

Dic54 in GAP, Magma, Sage, TeX

{\rm Dic}_{54}
% in TeX

G:=Group("Dic54");
// GroupNames label

G:=SmallGroup(216,4);
// by ID

G=gap.SmallGroup(216,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,963,381,3604,208,5189]);
// Polycyclic

G:=Group<a,b|a^108=1,b^2=a^54,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic54 in TeX

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