metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D90.C2, D45⋊2C4, C30.3D6, Dic9⋊2D5, Dic5⋊2D9, C10.3D18, C18.3D10, C90.3C22, C9⋊1(C4×D5), C5⋊2(C4×D9), C45⋊5(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
| C45 — D90.C2 |
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
3D30
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)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(115 180)(116 179)(117 178)(118 177)(119 176)(120 175)(121 174)(122 173)(123 172)(124 171)(125 170)(126 169)(127 168)(128 167)(129 166)(130 165)(131 164)(132 163)(133 162)(134 161)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)
(1 103 46 148)(2 122 47 167)(3 141 48 96)(4 160 49 115)(5 179 50 134)(6 108 51 153)(7 127 52 172)(8 146 53 101)(9 165 54 120)(10 94 55 139)(11 113 56 158)(12 132 57 177)(13 151 58 106)(14 170 59 125)(15 99 60 144)(16 118 61 163)(17 137 62 92)(18 156 63 111)(19 175 64 130)(20 104 65 149)(21 123 66 168)(22 142 67 97)(23 161 68 116)(24 180 69 135)(25 109 70 154)(26 128 71 173)(27 147 72 102)(28 166 73 121)(29 95 74 140)(30 114 75 159)(31 133 76 178)(32 152 77 107)(33 171 78 126)(34 100 79 145)(35 119 80 164)(36 138 81 93)(37 157 82 112)(38 176 83 131)(39 105 84 150)(40 124 85 169)(41 143 86 98)(42 162 87 117)(43 91 88 136)(44 110 89 155)(45 129 90 174)
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)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(115,180)(116,179)(117,178)(118,177)(119,176)(120,175)(121,174)(122,173)(123,172)(124,171)(125,170)(126,169)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148), (1,103,46,148)(2,122,47,167)(3,141,48,96)(4,160,49,115)(5,179,50,134)(6,108,51,153)(7,127,52,172)(8,146,53,101)(9,165,54,120)(10,94,55,139)(11,113,56,158)(12,132,57,177)(13,151,58,106)(14,170,59,125)(15,99,60,144)(16,118,61,163)(17,137,62,92)(18,156,63,111)(19,175,64,130)(20,104,65,149)(21,123,66,168)(22,142,67,97)(23,161,68,116)(24,180,69,135)(25,109,70,154)(26,128,71,173)(27,147,72,102)(28,166,73,121)(29,95,74,140)(30,114,75,159)(31,133,76,178)(32,152,77,107)(33,171,78,126)(34,100,79,145)(35,119,80,164)(36,138,81,93)(37,157,82,112)(38,176,83,131)(39,105,84,150)(40,124,85,169)(41,143,86,98)(42,162,87,117)(43,91,88,136)(44,110,89,155)(45,129,90,174)>;
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)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(115,180)(116,179)(117,178)(118,177)(119,176)(120,175)(121,174)(122,173)(123,172)(124,171)(125,170)(126,169)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148), (1,103,46,148)(2,122,47,167)(3,141,48,96)(4,160,49,115)(5,179,50,134)(6,108,51,153)(7,127,52,172)(8,146,53,101)(9,165,54,120)(10,94,55,139)(11,113,56,158)(12,132,57,177)(13,151,58,106)(14,170,59,125)(15,99,60,144)(16,118,61,163)(17,137,62,92)(18,156,63,111)(19,175,64,130)(20,104,65,149)(21,123,66,168)(22,142,67,97)(23,161,68,116)(24,180,69,135)(25,109,70,154)(26,128,71,173)(27,147,72,102)(28,166,73,121)(29,95,74,140)(30,114,75,159)(31,133,76,178)(32,152,77,107)(33,171,78,126)(34,100,79,145)(35,119,80,164)(36,138,81,93)(37,157,82,112)(38,176,83,131)(39,105,84,150)(40,124,85,169)(41,143,86,98)(42,162,87,117)(43,91,88,136)(44,110,89,155)(45,129,90,174) );
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),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(115,180),(116,179),(117,178),(118,177),(119,176),(120,175),(121,174),(122,173),(123,172),(124,171),(125,170),(126,169),(127,168),(128,167),(129,166),(130,165),(131,164),(132,163),(133,162),(134,161),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148)], [(1,103,46,148),(2,122,47,167),(3,141,48,96),(4,160,49,115),(5,179,50,134),(6,108,51,153),(7,127,52,172),(8,146,53,101),(9,165,54,120),(10,94,55,139),(11,113,56,158),(12,132,57,177),(13,151,58,106),(14,170,59,125),(15,99,60,144),(16,118,61,163),(17,137,62,92),(18,156,63,111),(19,175,64,130),(20,104,65,149),(21,123,66,168),(22,142,67,97),(23,161,68,116),(24,180,69,135),(25,109,70,154),(26,128,71,173),(27,147,72,102),(28,166,73,121),(29,95,74,140),(30,114,75,159),(31,133,76,178),(32,152,77,107),(33,171,78,126),(34,100,79,145),(35,119,80,164),(36,138,81,93),(37,157,82,112),(38,176,83,131),(39,105,84,150),(40,124,85,169),(41,143,86,98),(42,162,87,117),(43,91,88,136),(44,110,89,155),(45,129,90,174)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 9A | 9B | 9C | 10A | 10B | 12A | 12B | 15A | 15B | 18A | 18B | 18C | 20A | 20B | 20C | 20D | 30A | 30B | 36A | ··· | 36F | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 12 | 12 | 15 | 15 | 18 | 18 | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 45 | 45 | 2 | 5 | 5 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D9 | D10 | C4×S3 | D18 | C4×D5 | C4×D9 | S3×D5 | D30.C2 | D5×D9 | D90.C2 |
| kernel | D90.C2 | C5×Dic9 | C9×Dic5 | D90 | D45 | C3×Dic5 | Dic9 | C30 | Dic5 | C18 | C15 | C10 | C9 | C5 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 2 | 2 | 6 | 6 |
Matrix representation of D90.C2 ►in GL4(𝔽181) generated by
| 14 | 14 | 0 | 0 |
| 167 | 180 | 0 | 0 |
| 0 | 0 | 177 | 127 |
| 0 | 0 | 54 | 50 |
| 14 | 14 | 0 | 0 |
| 180 | 167 | 0 | 0 |
| 0 | 0 | 54 | 50 |
| 0 | 0 | 177 | 127 |
| 1 | 0 | 0 | 0 |
| 167 | 180 | 0 | 0 |
| 0 | 0 | 19 | 0 |
| 0 | 0 | 0 | 19 |
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