metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic45, C9⋊Dic5, C45⋊3C4, C2.D45, C18.D5, C10.D9, C5⋊2Dic9, C90.1C2, C30.1S3, C6.1D15, C3.Dic15, C15.2Dic3, SmallGroup(180,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — Dic45 |
Generators and relations for Dic45
G = < a,b | a90=1, b2=a45, bab-1=a-1 >
Smallest permutation representation of Dic45
►Regular action on 180 points
Generators in S180
(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)
(1 144 46 99)(2 143 47 98)(3 142 48 97)(4 141 49 96)(5 140 50 95)(6 139 51 94)(7 138 52 93)(8 137 53 92)(9 136 54 91)(10 135 55 180)(11 134 56 179)(12 133 57 178)(13 132 58 177)(14 131 59 176)(15 130 60 175)(16 129 61 174)(17 128 62 173)(18 127 63 172)(19 126 64 171)(20 125 65 170)(21 124 66 169)(22 123 67 168)(23 122 68 167)(24 121 69 166)(25 120 70 165)(26 119 71 164)(27 118 72 163)(28 117 73 162)(29 116 74 161)(30 115 75 160)(31 114 76 159)(32 113 77 158)(33 112 78 157)(34 111 79 156)(35 110 80 155)(36 109 81 154)(37 108 82 153)(38 107 83 152)(39 106 84 151)(40 105 85 150)(41 104 86 149)(42 103 87 148)(43 102 88 147)(44 101 89 146)(45 100 90 145)
G:=sub<Sym(180)| (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), (1,144,46,99)(2,143,47,98)(3,142,48,97)(4,141,49,96)(5,140,50,95)(6,139,51,94)(7,138,52,93)(8,137,53,92)(9,136,54,91)(10,135,55,180)(11,134,56,179)(12,133,57,178)(13,132,58,177)(14,131,59,176)(15,130,60,175)(16,129,61,174)(17,128,62,173)(18,127,63,172)(19,126,64,171)(20,125,65,170)(21,124,66,169)(22,123,67,168)(23,122,68,167)(24,121,69,166)(25,120,70,165)(26,119,71,164)(27,118,72,163)(28,117,73,162)(29,116,74,161)(30,115,75,160)(31,114,76,159)(32,113,77,158)(33,112,78,157)(34,111,79,156)(35,110,80,155)(36,109,81,154)(37,108,82,153)(38,107,83,152)(39,106,84,151)(40,105,85,150)(41,104,86,149)(42,103,87,148)(43,102,88,147)(44,101,89,146)(45,100,90,145)>;
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), (1,144,46,99)(2,143,47,98)(3,142,48,97)(4,141,49,96)(5,140,50,95)(6,139,51,94)(7,138,52,93)(8,137,53,92)(9,136,54,91)(10,135,55,180)(11,134,56,179)(12,133,57,178)(13,132,58,177)(14,131,59,176)(15,130,60,175)(16,129,61,174)(17,128,62,173)(18,127,63,172)(19,126,64,171)(20,125,65,170)(21,124,66,169)(22,123,67,168)(23,122,68,167)(24,121,69,166)(25,120,70,165)(26,119,71,164)(27,118,72,163)(28,117,73,162)(29,116,74,161)(30,115,75,160)(31,114,76,159)(32,113,77,158)(33,112,78,157)(34,111,79,156)(35,110,80,155)(36,109,81,154)(37,108,82,153)(38,107,83,152)(39,106,84,151)(40,105,85,150)(41,104,86,149)(42,103,87,148)(43,102,88,147)(44,101,89,146)(45,100,90,145) );
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)], [(1,144,46,99),(2,143,47,98),(3,142,48,97),(4,141,49,96),(5,140,50,95),(6,139,51,94),(7,138,52,93),(8,137,53,92),(9,136,54,91),(10,135,55,180),(11,134,56,179),(12,133,57,178),(13,132,58,177),(14,131,59,176),(15,130,60,175),(16,129,61,174),(17,128,62,173),(18,127,63,172),(19,126,64,171),(20,125,65,170),(21,124,66,169),(22,123,67,168),(23,122,68,167),(24,121,69,166),(25,120,70,165),(26,119,71,164),(27,118,72,163),(28,117,73,162),(29,116,74,161),(30,115,75,160),(31,114,76,159),(32,113,77,158),(33,112,78,157),(34,111,79,156),(35,110,80,155),(36,109,81,154),(37,108,82,153),(38,107,83,152),(39,106,84,151),(40,105,85,150),(41,104,86,149),(42,103,87,148),(43,102,88,147),(44,101,89,146),(45,100,90,145)]])
Dic45 is a maximal subgroup of
C45⋊Q8 D9×Dic5 D5×Dic9 C45⋊D4 Dic90 C4×D45 C45⋊7D4
Dic45 is a maximal quotient of C45⋊3C8
48 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 9A | 9B | 9C | 10A | 10B | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 30A | 30B | 30C | 30D | 45A | ··· | 45L | 90A | ··· | 90L |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 30 | 30 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | - | + | - | |
| image | C1 | C2 | C4 | S3 | D5 | Dic3 | D9 | Dic5 | D15 | Dic9 | Dic15 | D45 | Dic45 |
| kernel | Dic45 | C90 | C45 | C30 | C18 | C15 | C10 | C9 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 2 | 4 | 3 | 4 | 12 | 12 |
Matrix representation of Dic45 ►in GL3(𝔽181) generated by
| 180 | 0 | 0 |
| 0 | 41 | 93 |
| 0 | 88 | 129 |
| 19 | 0 | 0 |
| 0 | 106 | 62 |
| 0 | 137 | 75 |
G:=sub<GL(3,GF(181))| [180,0,0,0,41,88,0,93,129],[19,0,0,0,106,137,0,62,75] >;
Dic45 in GAP, Magma, Sage, TeX
{\rm Dic}_{45} % in TeX
G:=Group("Dic45"); // GroupNames label
G:=SmallGroup(180,3);
// by ID
G=gap.SmallGroup(180,3);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-3,10,1022,462,963,3004]);
// Polycyclic
G:=Group<a,b|a^90=1,b^2=a^45,b*a*b^-1=a^-1>;
// generators/relations
Export