metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic55, C55⋊3C4, C2.D55, C22.D5, C11⋊Dic5, C10.D11, C5⋊2Dic11, C110.1C2, SmallGroup(220,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — Dic55 |
Generators and relations for Dic55
G = < a,b | a110=1, b2=a55, bab-1=a-1 >
Smallest permutation representation of Dic55
►Regular action on 220 points
Generators in S220
(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)
(1 216 56 161)(2 215 57 160)(3 214 58 159)(4 213 59 158)(5 212 60 157)(6 211 61 156)(7 210 62 155)(8 209 63 154)(9 208 64 153)(10 207 65 152)(11 206 66 151)(12 205 67 150)(13 204 68 149)(14 203 69 148)(15 202 70 147)(16 201 71 146)(17 200 72 145)(18 199 73 144)(19 198 74 143)(20 197 75 142)(21 196 76 141)(22 195 77 140)(23 194 78 139)(24 193 79 138)(25 192 80 137)(26 191 81 136)(27 190 82 135)(28 189 83 134)(29 188 84 133)(30 187 85 132)(31 186 86 131)(32 185 87 130)(33 184 88 129)(34 183 89 128)(35 182 90 127)(36 181 91 126)(37 180 92 125)(38 179 93 124)(39 178 94 123)(40 177 95 122)(41 176 96 121)(42 175 97 120)(43 174 98 119)(44 173 99 118)(45 172 100 117)(46 171 101 116)(47 170 102 115)(48 169 103 114)(49 168 104 113)(50 167 105 112)(51 166 106 111)(52 165 107 220)(53 164 108 219)(54 163 109 218)(55 162 110 217)
G:=sub<Sym(220)| (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), (1,216,56,161)(2,215,57,160)(3,214,58,159)(4,213,59,158)(5,212,60,157)(6,211,61,156)(7,210,62,155)(8,209,63,154)(9,208,64,153)(10,207,65,152)(11,206,66,151)(12,205,67,150)(13,204,68,149)(14,203,69,148)(15,202,70,147)(16,201,71,146)(17,200,72,145)(18,199,73,144)(19,198,74,143)(20,197,75,142)(21,196,76,141)(22,195,77,140)(23,194,78,139)(24,193,79,138)(25,192,80,137)(26,191,81,136)(27,190,82,135)(28,189,83,134)(29,188,84,133)(30,187,85,132)(31,186,86,131)(32,185,87,130)(33,184,88,129)(34,183,89,128)(35,182,90,127)(36,181,91,126)(37,180,92,125)(38,179,93,124)(39,178,94,123)(40,177,95,122)(41,176,96,121)(42,175,97,120)(43,174,98,119)(44,173,99,118)(45,172,100,117)(46,171,101,116)(47,170,102,115)(48,169,103,114)(49,168,104,113)(50,167,105,112)(51,166,106,111)(52,165,107,220)(53,164,108,219)(54,163,109,218)(55,162,110,217)>;
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), (1,216,56,161)(2,215,57,160)(3,214,58,159)(4,213,59,158)(5,212,60,157)(6,211,61,156)(7,210,62,155)(8,209,63,154)(9,208,64,153)(10,207,65,152)(11,206,66,151)(12,205,67,150)(13,204,68,149)(14,203,69,148)(15,202,70,147)(16,201,71,146)(17,200,72,145)(18,199,73,144)(19,198,74,143)(20,197,75,142)(21,196,76,141)(22,195,77,140)(23,194,78,139)(24,193,79,138)(25,192,80,137)(26,191,81,136)(27,190,82,135)(28,189,83,134)(29,188,84,133)(30,187,85,132)(31,186,86,131)(32,185,87,130)(33,184,88,129)(34,183,89,128)(35,182,90,127)(36,181,91,126)(37,180,92,125)(38,179,93,124)(39,178,94,123)(40,177,95,122)(41,176,96,121)(42,175,97,120)(43,174,98,119)(44,173,99,118)(45,172,100,117)(46,171,101,116)(47,170,102,115)(48,169,103,114)(49,168,104,113)(50,167,105,112)(51,166,106,111)(52,165,107,220)(53,164,108,219)(54,163,109,218)(55,162,110,217) );
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)], [(1,216,56,161),(2,215,57,160),(3,214,58,159),(4,213,59,158),(5,212,60,157),(6,211,61,156),(7,210,62,155),(8,209,63,154),(9,208,64,153),(10,207,65,152),(11,206,66,151),(12,205,67,150),(13,204,68,149),(14,203,69,148),(15,202,70,147),(16,201,71,146),(17,200,72,145),(18,199,73,144),(19,198,74,143),(20,197,75,142),(21,196,76,141),(22,195,77,140),(23,194,78,139),(24,193,79,138),(25,192,80,137),(26,191,81,136),(27,190,82,135),(28,189,83,134),(29,188,84,133),(30,187,85,132),(31,186,86,131),(32,185,87,130),(33,184,88,129),(34,183,89,128),(35,182,90,127),(36,181,91,126),(37,180,92,125),(38,179,93,124),(39,178,94,123),(40,177,95,122),(41,176,96,121),(42,175,97,120),(43,174,98,119),(44,173,99,118),(45,172,100,117),(46,171,101,116),(47,170,102,115),(48,169,103,114),(49,168,104,113),(50,167,105,112),(51,166,106,111),(52,165,107,220),(53,164,108,219),(54,163,109,218),(55,162,110,217)]])
Dic55 is a maximal subgroup of
Dic5×D11 D5×Dic11 C55⋊D4 C55⋊Q8 Dic110 C4×D55 C55⋊7D4
Dic55 is a maximal quotient of C55⋊3C8
58 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 22A | ··· | 22E | 55A | ··· | 55T | 110A | ··· | 110T |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 55 | 55 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D5 | Dic5 | D11 | Dic11 | D55 | Dic55 |
| kernel | Dic55 | C110 | C55 | C22 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 20 | 20 |
Matrix representation of Dic55 ►in GL2(𝔽661) generated by
| 153 | 219 |
| 442 | 84 |
| 334 | 265 |
| 33 | 327 |
G:=sub<GL(2,GF(661))| [153,442,219,84],[334,33,265,327] >;
Dic55 in GAP, Magma, Sage, TeX
{\rm Dic}_{55} % in TeX
G:=Group("Dic55"); // GroupNames label
G:=SmallGroup(220,5);
// by ID
G=gap.SmallGroup(220,5);
# by ID
G:=PCGroup([4,-2,-2,-5,-11,8,194,3203]);
// Polycyclic
G:=Group<a,b|a^110=1,b^2=a^55,b*a*b^-1=a^-1>;
// generators/relations
Export