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G = D9×C20order 360 = 23·32·5

Direct product of C20 and D9

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D9×C20, C1805C2, C362C10, D18.C10, C60.13S3, C30.57D6, Dic92C10, C10.14D18, C90.14C22, C91(C2×C20), C458(C2×C4), C3.(S3×C20), C15.5(C4×S3), C6.7(S3×C10), C12.5(C5×S3), C2.1(C10×D9), C18.2(C2×C10), (C5×Dic9)⋊5C2, (C10×D9).2C2, SmallGroup(360,21)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C20
C1C3C9C18C90C10×D9 — D9×C20
C9 — D9×C20
C1C20

Generators and relations for D9×C20
 G = < a,b,c | a20=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
9C4
9C22
3S3
3S3
9C10
9C10
9C2×C4
3Dic3
3D6
9C20
9C2×C10
3C5×S3
3C5×S3
3C4×S3
9C2×C20
3S3×C10
3C5×Dic3
3S3×C20

Smallest permutation representation of D9×C20
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)
(1 113 29 124 160 70 173 96 43)(2 114 30 125 141 71 174 97 44)(3 115 31 126 142 72 175 98 45)(4 116 32 127 143 73 176 99 46)(5 117 33 128 144 74 177 100 47)(6 118 34 129 145 75 178 81 48)(7 119 35 130 146 76 179 82 49)(8 120 36 131 147 77 180 83 50)(9 101 37 132 148 78 161 84 51)(10 102 38 133 149 79 162 85 52)(11 103 39 134 150 80 163 86 53)(12 104 40 135 151 61 164 87 54)(13 105 21 136 152 62 165 88 55)(14 106 22 137 153 63 166 89 56)(15 107 23 138 154 64 167 90 57)(16 108 24 139 155 65 168 91 58)(17 109 25 140 156 66 169 92 59)(18 110 26 121 157 67 170 93 60)(19 111 27 122 158 68 171 94 41)(20 112 28 123 159 69 172 95 42)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 175)(22 176)(23 177)(24 178)(25 179)(26 180)(27 161)(28 162)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(61 125)(62 126)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 121)(78 122)(79 123)(80 124)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B5C5D 6 9A9B9C10A10B10C10D10E···10L12A12B15A15B15C15D18A18B18C20A···20H20I···20P30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order122234444555569991010101010···1012121515151518181820···2020···203030303036···3645···4560···6090···90180···180
size1199211991111222211119···92222222221···19···922222···22···22···22···22···2

120 irreducible representations

dim1111111111222222222222
type++++++++
imageC1C2C2C2C4C5C10C10C10C20S3D6D9C4×S3C5×S3D18S3×C10C4×D9C5×D9S3×C20C10×D9D9×C20
kernelD9×C20C5×Dic9C180C10×D9C5×D9C4×D9Dic9C36D18D9C60C30C20C15C12C10C6C5C4C3C2C1
# reps11114444416113243461281224

Matrix representation of D9×C20 in GL3(𝔽181) generated by

13900
01620
00162
,
100
0177131
050127
,
18000
054177
050127
G:=sub<GL(3,GF(181))| [139,0,0,0,162,0,0,0,162],[1,0,0,0,177,50,0,131,127],[180,0,0,0,54,50,0,177,127] >;

D9×C20 in GAP, Magma, Sage, TeX

D_9\times C_{20}
% in TeX

G:=Group("D9xC20");
// GroupNames label

G:=SmallGroup(360,21);
// by ID

G=gap.SmallGroup(360,21);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,6004,208,8645]);
// Polycyclic

G:=Group<a,b,c|a^20=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D9×C20 in TeX

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