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## G = C9×C5⋊D4order 360 = 23·32·5

### Direct product of C9 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C9×C5⋊D4
 Chief series C1 — C5 — C15 — C30 — C90 — D5×C18 — C9×C5⋊D4
 Lower central C5 — C10 — C9×C5⋊D4
 Upper central C1 — C18 — C2×C18

Generators and relations for C9×C5⋊D4
G = < a,b,c,d | a9=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C9×C5⋊D4
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)
(1 161 134 22 88)(2 162 135 23 89)(3 154 127 24 90)(4 155 128 25 82)(5 156 129 26 83)(6 157 130 27 84)(7 158 131 19 85)(8 159 132 20 86)(9 160 133 21 87)(10 65 41 171 124)(11 66 42 163 125)(12 67 43 164 126)(13 68 44 165 118)(14 69 45 166 119)(15 70 37 167 120)(16 71 38 168 121)(17 72 39 169 122)(18 64 40 170 123)(28 142 177 153 59)(29 143 178 145 60)(30 144 179 146 61)(31 136 180 147 62)(32 137 172 148 63)(33 138 173 149 55)(34 139 174 150 56)(35 140 175 151 57)(36 141 176 152 58)(46 75 91 113 101)(47 76 92 114 102)(48 77 93 115 103)(49 78 94 116 104)(50 79 95 117 105)(51 80 96 109 106)(52 81 97 110 107)(53 73 98 111 108)(54 74 99 112 100)
(1 145 102 37)(2 146 103 38)(3 147 104 39)(4 148 105 40)(5 149 106 41)(6 150 107 42)(7 151 108 43)(8 152 100 44)(9 153 101 45)(10 129 138 80)(11 130 139 81)(12 131 140 73)(13 132 141 74)(14 133 142 75)(15 134 143 76)(16 135 144 77)(17 127 136 78)(18 128 137 79)(19 35 98 126)(20 36 99 118)(21 28 91 119)(22 29 92 120)(23 30 93 121)(24 31 94 122)(25 32 95 123)(26 33 96 124)(27 34 97 125)(46 69 160 177)(47 70 161 178)(48 71 162 179)(49 72 154 180)(50 64 155 172)(51 65 156 173)(52 66 157 174)(53 67 158 175)(54 68 159 176)(55 109 171 83)(56 110 163 84)(57 111 164 85)(58 112 165 86)(59 113 166 87)(60 114 167 88)(61 115 168 89)(62 116 169 90)(63 117 170 82)
(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)(19 131)(20 132)(21 133)(22 134)(23 135)(24 127)(25 128)(26 129)(27 130)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 113)(47 114)(48 115)(49 116)(50 117)(51 109)(52 110)(53 111)(54 112)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 64)(73 98)(74 99)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 154)(118 141)(119 142)(120 143)(121 144)(122 136)(123 137)(124 138)(125 139)(126 140)(163 174)(164 175)(165 176)(166 177)(167 178)(168 179)(169 180)(170 172)(171 173)

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

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

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

117 conjugacy classes

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

117 irreducible representations

 dim 1 1 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 C3 C6 C6 C6 C9 C18 C18 C18 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 D4×C9 C9×D5 C3×C5⋊D4 D5×C18 C9×C5⋊D4 kernel C9×C5⋊D4 C9×Dic5 D5×C18 C2×C90 C3×C5⋊D4 C3×Dic5 C6×D5 C2×C30 C5⋊D4 Dic5 D10 C2×C10 C45 C2×C18 C18 C15 C2×C6 C9 C6 C5 C22 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 2 2 2 4 4 4 6 12 8 12 24

Matrix representation of C9×C5⋊D4 in GL2(𝔽181) generated by

 80 0 0 80
,
 167 180 1 0
,
 155 48 50 26
,
 1 0 167 180
G:=sub<GL(2,GF(181))| [80,0,0,80],[167,1,180,0],[155,50,48,26],[1,167,0,180] >;

C9×C5⋊D4 in GAP, Magma, Sage, TeX

C_9\times C_5\rtimes D_4
% in TeX

G:=Group("C9xC5:D4");
// GroupNames label

G:=SmallGroup(360,19);
// by ID

G=gap.SmallGroup(360,19);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,122,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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