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## G = D9×C20order 360 = 23·32·5

### Direct product of C20 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C20
 Chief series C1 — C3 — C9 — C18 — C90 — C10×D9 — D9×C20
 Lower central C9 — D9×C20
 Upper central C1 — C20

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

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 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6 9A 9B 9C 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 18A 18B 18C 20A ··· 20H 20I ··· 20P 30A 30B 30C 30D 36A ··· 36F 45A ··· 45L 60A ··· 60H 90A ··· 90L 180A ··· 180X order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 9 9 9 10 10 10 10 10 ··· 10 12 12 15 15 15 15 18 18 18 20 ··· 20 20 ··· 20 30 30 30 30 36 ··· 36 45 ··· 45 60 ··· 60 90 ··· 90 180 ··· 180 size 1 1 9 9 2 1 1 9 9 1 1 1 1 2 2 2 2 1 1 1 1 9 ··· 9 2 2 2 2 2 2 2 2 2 1 ··· 1 9 ··· 9 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 D6 D9 C4×S3 C5×S3 D18 S3×C10 C4×D9 C5×D9 S3×C20 C10×D9 D9×C20 kernel D9×C20 C5×Dic9 C180 C10×D9 C5×D9 C4×D9 Dic9 C36 D18 D9 C60 C30 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 3 2 4 3 4 6 12 8 12 24

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

 139 0 0 0 162 0 0 0 162
,
 1 0 0 0 177 131 0 50 127
,
 180 0 0 0 54 177 0 50 127
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

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