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## G = C5×Dic9order 180 = 22·32·5

### Direct product of C5 and Dic9

Aliases: C5×Dic9, C9⋊C20, C454C4, C18.C10, C90.2C2, C30.5S3, C10.2D9, C15.3Dic3, C2.(C5×D9), C6.1(C5×S3), C3.(C5×Dic3), SmallGroup(180,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C5×Dic9
 Chief series C1 — C3 — C9 — C18 — C90 — C5×Dic9
 Lower central C9 — C5×Dic9
 Upper central C1 — C10

Generators and relations for C5×Dic9
G = < a,b,c | a5=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic9
Regular action on 180 points
Generators in S180
(1 73 62 47 29)(2 74 63 48 30)(3 75 64 49 31)(4 76 65 50 32)(5 77 66 51 33)(6 78 67 52 34)(7 79 68 53 35)(8 80 69 54 36)(9 81 70 37 19)(10 82 71 38 20)(11 83 72 39 21)(12 84 55 40 22)(13 85 56 41 23)(14 86 57 42 24)(15 87 58 43 25)(16 88 59 44 26)(17 89 60 45 27)(18 90 61 46 28)(91 171 145 144 116)(92 172 146 127 117)(93 173 147 128 118)(94 174 148 129 119)(95 175 149 130 120)(96 176 150 131 121)(97 177 151 132 122)(98 178 152 133 123)(99 179 153 134 124)(100 180 154 135 125)(101 163 155 136 126)(102 164 156 137 109)(103 165 157 138 110)(104 166 158 139 111)(105 167 159 140 112)(106 168 160 141 113)(107 169 161 142 114)(108 170 162 143 115)
(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)
(1 100 10 91)(2 99 11 108)(3 98 12 107)(4 97 13 106)(5 96 14 105)(6 95 15 104)(7 94 16 103)(8 93 17 102)(9 92 18 101)(19 117 28 126)(20 116 29 125)(21 115 30 124)(22 114 31 123)(23 113 32 122)(24 112 33 121)(25 111 34 120)(26 110 35 119)(27 109 36 118)(37 127 46 136)(38 144 47 135)(39 143 48 134)(40 142 49 133)(41 141 50 132)(42 140 51 131)(43 139 52 130)(44 138 53 129)(45 137 54 128)(55 161 64 152)(56 160 65 151)(57 159 66 150)(58 158 67 149)(59 157 68 148)(60 156 69 147)(61 155 70 146)(62 154 71 145)(63 153 72 162)(73 180 82 171)(74 179 83 170)(75 178 84 169)(76 177 85 168)(77 176 86 167)(78 175 87 166)(79 174 88 165)(80 173 89 164)(81 172 90 163)

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

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

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

C5×Dic9 is a maximal subgroup of   C45⋊Q8  D90.C2  C9⋊D20  D9×C20

60 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C 5D 6 9A 9B 9C 10A 10B 10C 10D 15A 15B 15C 15D 18A 18B 18C 20A ··· 20H 30A 30B 30C 30D 45A ··· 45L 90A ··· 90L order 1 2 3 4 4 5 5 5 5 6 9 9 9 10 10 10 10 15 15 15 15 18 18 18 20 ··· 20 30 30 30 30 45 ··· 45 90 ··· 90 size 1 1 2 9 9 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 9 ··· 9 2 2 2 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + - image C1 C2 C4 C5 C10 C20 S3 Dic3 D9 C5×S3 Dic9 C5×Dic3 C5×D9 C5×Dic9 kernel C5×Dic9 C90 C45 Dic9 C18 C9 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 4 4 8 1 1 3 4 3 4 12 12

Matrix representation of C5×Dic9 in GL3(𝔽181) generated by

 59 0 0 0 42 0 0 0 42
,
 180 0 0 0 4 54 0 127 131
,
 19 0 0 0 131 4 0 54 50
G:=sub<GL(3,GF(181))| [59,0,0,0,42,0,0,0,42],[180,0,0,0,4,127,0,54,131],[19,0,0,0,131,54,0,4,50] >;

C5×Dic9 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_9
% in TeX

G:=Group("C5xDic9");
// GroupNames label

G:=SmallGroup(180,1);
// by ID

G=gap.SmallGroup(180,1);
# by ID

G:=PCGroup([5,-2,-5,-2,-3,-3,50,2003,138,3004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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