Copied to
clipboard

G = D90.C2order 360 = 23·32·5

The non-split extension by D90 of C2 acting faithfully

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D90.C2, D452C4, C30.3D6, Dic92D5, Dic52D9, C10.3D18, C18.3D10, C90.3C22, C91(C4×D5), C52(C4×D9), C455(C2×C4), C2.2(D5×D9), C15.3(C4×S3), C6.10(S3×D5), (C5×Dic9)⋊1C2, (C9×Dic5)⋊2C2, C3.(D30.C2), (C3×Dic5).3S3, SmallGroup(360,9)

Series: Derived Chief Lower central Upper central

C1C45 — D90.C2
C1C3C15C45C90C9×Dic5 — D90.C2
C45 — D90.C2
C1C2

Generators and relations for D90.C2
 G = < a,b,c | a90=b2=1, c2=a45, bab=a-1, cac-1=a19, cbc-1=a18b >

45C2
45C2
5C4
9C4
45C22
15S3
15S3
9D5
9D5
45C2×C4
3Dic3
5C12
15D6
5D9
5D9
9C20
9D10
3D15
3D15
15C4×S3
5C36
5D18
9C4×D5
3C5×Dic3
3D30
5C4×D9
3D30.C2

Smallest permutation representation of D90.C2
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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(91 94)(92 93)(95 180)(96 179)(97 178)(98 177)(99 176)(100 175)(101 174)(102 173)(103 172)(104 171)(105 170)(106 169)(107 168)(108 167)(109 166)(110 165)(111 164)(112 163)(113 162)(114 161)(115 160)(116 159)(117 158)(118 157)(119 156)(120 155)(121 154)(122 153)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)
(1 93 46 138)(2 112 47 157)(3 131 48 176)(4 150 49 105)(5 169 50 124)(6 98 51 143)(7 117 52 162)(8 136 53 91)(9 155 54 110)(10 174 55 129)(11 103 56 148)(12 122 57 167)(13 141 58 96)(14 160 59 115)(15 179 60 134)(16 108 61 153)(17 127 62 172)(18 146 63 101)(19 165 64 120)(20 94 65 139)(21 113 66 158)(22 132 67 177)(23 151 68 106)(24 170 69 125)(25 99 70 144)(26 118 71 163)(27 137 72 92)(28 156 73 111)(29 175 74 130)(30 104 75 149)(31 123 76 168)(32 142 77 97)(33 161 78 116)(34 180 79 135)(35 109 80 154)(36 128 81 173)(37 147 82 102)(38 166 83 121)(39 95 84 140)(40 114 85 159)(41 133 86 178)(42 152 87 107)(43 171 88 126)(44 100 89 145)(45 119 90 164)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 9A9B9C10A10B12A12B15A15B18A18B18C20A20B20C20D30A30B36A···36F45A···45F90A···90F
order12223444455699910101212151518181820202020303036···3645···4590···90
size1145452559922222222101044222181818184410···104···44···4

48 irreducible representations

dim111112222222224444
type++++++++++++++
imageC1C2C2C2C4S3D5D6D9D10C4×S3D18C4×D5C4×D9S3×D5D30.C2D5×D9D90.C2
kernelD90.C2C5×Dic9C9×Dic5D90D45C3×Dic5Dic9C30Dic5C18C15C10C9C5C6C3C2C1
# reps111141213223462266

Matrix representation of D90.C2 in GL4(𝔽181) generated by

141400
16718000
00177127
005450
,
141400
18016700
005450
00177127
,
1000
16718000
00190
00019
G:=sub<GL(4,GF(181))| [14,167,0,0,14,180,0,0,0,0,177,54,0,0,127,50],[14,180,0,0,14,167,0,0,0,0,54,177,0,0,50,127],[1,167,0,0,0,180,0,0,0,0,19,0,0,0,0,19] >;

D90.C2 in GAP, Magma, Sage, TeX

D_{90}.C_2
% in TeX

G:=Group("D90.C2");
// GroupNames label

G:=SmallGroup(360,9);
// by ID

G=gap.SmallGroup(360,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,31,1641,741,2884,4331]);
// Polycyclic

G:=Group<a,b,c|a^90=b^2=1,c^2=a^45,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations

Export

Subgroup lattice of D90.C2 in TeX

׿
×
𝔽