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## G = C9×Dic5order 180 = 22·32·5

### Direct product of C9 and Dic5

Aliases: C9×Dic5, C455C4, C52C36, C10.C18, C90.3C2, C30.2C6, C18.2D5, C15.2C12, C2.(C9×D5), C6.2(C3×D5), C3.(C3×Dic5), (C3×Dic5).C3, SmallGroup(180,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C9×Dic5
 Chief series C1 — C5 — C15 — C30 — C90 — C9×Dic5
 Lower central C5 — C9×Dic5
 Upper central C1 — C18

Generators and relations for C9×Dic5
G = < a,b,c | a9=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×Dic5
Regular action on 180 points
Generators in S180
(1 100 63 33 78 48 27 89 59)(2 91 64 34 79 49 28 90 60)(3 92 65 35 80 50 29 81 51)(4 93 66 36 71 41 30 82 52)(5 94 67 37 72 42 21 83 53)(6 95 68 38 73 43 22 84 54)(7 96 69 39 74 44 23 85 55)(8 97 70 40 75 45 24 86 56)(9 98 61 31 76 46 25 87 57)(10 99 62 32 77 47 26 88 58)(11 152 122 167 137 107 171 141 111)(12 153 123 168 138 108 172 142 112)(13 154 124 169 139 109 173 143 113)(14 155 125 170 140 110 174 144 114)(15 156 126 161 131 101 175 145 115)(16 157 127 162 132 102 176 146 116)(17 158 128 163 133 103 177 147 117)(18 159 129 164 134 104 178 148 118)(19 160 130 165 135 105 179 149 119)(20 151 121 166 136 106 180 150 120)
(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 100 16 95)(12 99 17 94)(13 98 18 93)(14 97 19 92)(15 96 20 91)(21 108 26 103)(22 107 27 102)(23 106 28 101)(24 105 29 110)(25 104 30 109)(31 129 36 124)(32 128 37 123)(33 127 38 122)(34 126 39 121)(35 125 40 130)(41 139 46 134)(42 138 47 133)(43 137 48 132)(44 136 49 131)(45 135 50 140)(51 144 56 149)(52 143 57 148)(53 142 58 147)(54 141 59 146)(55 150 60 145)(61 159 66 154)(62 158 67 153)(63 157 68 152)(64 156 69 151)(65 155 70 160)(71 169 76 164)(72 168 77 163)(73 167 78 162)(74 166 79 161)(75 165 80 170)(81 174 86 179)(82 173 87 178)(83 172 88 177)(84 171 89 176)(85 180 90 175)

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

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

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

C9×Dic5 is a maximal subgroup of   C45⋊C8  C45⋊Q8  D90.C2  C5⋊D36  D5×C36

72 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 9A ··· 9F 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 18A ··· 18F 30A 30B 30C 30D 36A ··· 36L 45A ··· 45L 90A ··· 90L order 1 2 3 3 4 4 5 5 6 6 9 ··· 9 10 10 12 12 12 12 15 15 15 15 18 ··· 18 30 30 30 30 36 ··· 36 45 ··· 45 90 ··· 90 size 1 1 1 1 5 5 2 2 1 1 1 ··· 1 2 2 5 5 5 5 2 2 2 2 1 ··· 1 2 2 2 2 5 ··· 5 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C9 C12 C18 C36 D5 Dic5 C3×D5 C3×Dic5 C9×D5 C9×Dic5 kernel C9×Dic5 C90 C3×Dic5 C45 C30 Dic5 C15 C10 C5 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 2 2 4 4 12 12

Matrix representation of C9×Dic5 in GL2(𝔽19) generated by

 5 0 0 5
,
 11 11 6 13
,
 0 10 17 0
G:=sub<GL(2,GF(19))| [5,0,0,5],[11,6,11,13],[0,17,10,0] >;

C9×Dic5 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_5
% in TeX

G:=Group("C9xDic5");
// GroupNames label

G:=SmallGroup(180,2);
// by ID

G=gap.SmallGroup(180,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,66,3604]);
// Polycyclic

G:=Group<a,b,c|a^9=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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