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
Generators and relations for Dic54
G = < a,b | a108=1, b2=a54, bab-1=a-1 >
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 D108⋊5C2 D4⋊2D27 Q8×D27
Dic54 is a maximal quotient of Dic27⋊C4 C4⋊Dic27
57 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 108A | ··· | 108R |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | D9 | Dic6 | D18 | D27 | Dic18 | D54 | Dic54 |
| kernel | Dic54 | Dic27 | C108 | C36 | C27 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 9 | 6 | 9 | 18 |
Matrix representation of Dic54 ►in GL2(𝔽109) generated by
| 60 | 104 |
| 5 | 65 |
| 45 | 20 |
| 84 | 64 |
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
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