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G = Dic45order 180 = 22·32·5

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic45, C9⋊Dic5, C453C4, C2.D45, C18.D5, C10.D9, C52Dic9, C90.1C2, C30.1S3, C6.1D15, C3.Dic15, C15.2Dic3, SmallGroup(180,3)

Series: Derived Chief Lower central Upper central

C1C45 — Dic45
C1C3C15C45C90 — Dic45
C45 — Dic45
C1C2

Generators and relations for Dic45
 G = < a,b | a90=1, b2=a45, bab-1=a-1 >

45C4
15Dic3
9Dic5
5Dic9
3Dic15

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

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

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

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

Dic45 is a maximal subgroup of   C45⋊Q8  D9×Dic5  D5×Dic9  C45⋊D4  Dic90  C4×D45  C457D4
Dic45 is a maximal quotient of   C453C8

48 conjugacy classes

class 1  2  3 4A4B5A5B 6 9A9B9C10A10B15A15B15C15D18A18B18C30A30B30C30D45A···45L90A···90L
order123445569991010151515151818183030303045···4590···90
size112454522222222222222222222···22···2

48 irreducible representations

dim1112222222222
type++++-+-+--+-
imageC1C2C4S3D5Dic3D9Dic5D15Dic9Dic15D45Dic45
kernelDic45C90C45C30C18C15C10C9C6C5C3C2C1
# reps112121324341212

Matrix representation of Dic45 in GL3(𝔽181) generated by

18000
04193
088129
,
1900
010662
013775
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

Subgroup lattice of Dic45 in TeX

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