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

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

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

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

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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