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## G = D9×C2×C10order 360 = 23·32·5

### Direct product of C2×C10 and D9

Aliases: D9×C2×C10, C453C23, C903C22, C30.61D6, C18⋊(C2×C10), C9⋊(C22×C10), (C2×C90)⋊5C2, (C2×C18)⋊3C10, C6.11(S3×C10), (C2×C30).10S3, C15.3(C22×S3), C3.(S3×C2×C10), (C2×C6).4(C5×S3), SmallGroup(360,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C2×C10
 Chief series C1 — C3 — C9 — C45 — C5×D9 — C10×D9 — D9×C2×C10
 Lower central C9 — D9×C2×C10
 Upper central C1 — C2×C10

Generators and relations for D9×C2×C10
G = < a,b,c,d | a2=b10=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 96 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C9, C10, C10, D6, C2×C6, C15, D9, C18, C2×C10, C2×C10, C22×S3, C5×S3, C30, D18, C2×C18, C22×C10, C45, S3×C10, C2×C30, C22×D9, C5×D9, C90, S3×C2×C10, C10×D9, C2×C90, D9×C2×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, D9, C2×C10, C22×S3, C5×S3, D18, C22×C10, S3×C10, C22×D9, C5×D9, S3×C2×C10, C10×D9, D9×C2×C10

Smallest permutation representation of D9×C2×C10
On 180 points
Generators in S180
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 171)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 151)(18 152)(19 153)(20 154)(21 165)(22 166)(23 167)(24 168)(25 169)(26 170)(27 161)(28 162)(29 163)(30 164)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 91)(40 92)(41 140)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 150)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 101)(68 102)(69 103)(70 104)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 111)(78 112)(79 113)(80 114)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 121)(88 122)(89 123)(90 124)
(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 98 141 157 118 133 122 101 167)(2 99 142 158 119 134 123 102 168)(3 100 143 159 120 135 124 103 169)(4 91 144 160 111 136 125 104 170)(5 92 145 151 112 137 126 105 161)(6 93 146 152 113 138 127 106 162)(7 94 147 153 114 139 128 107 163)(8 95 148 154 115 140 129 108 164)(9 96 149 155 116 131 130 109 165)(10 97 150 156 117 132 121 110 166)(11 72 42 86 65 21 180 34 60)(12 73 43 87 66 22 171 35 51)(13 74 44 88 67 23 172 36 52)(14 75 45 89 68 24 173 37 53)(15 76 46 90 69 25 174 38 54)(16 77 47 81 70 26 175 39 55)(17 78 48 82 61 27 176 40 56)(18 79 49 83 62 28 177 31 57)(19 80 50 84 63 29 178 32 58)(20 71 41 85 64 30 179 33 59)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 136)(12 137)(13 138)(14 139)(15 140)(16 131)(17 132)(18 133)(19 134)(20 135)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 159)(42 160)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 97)(62 98)(63 99)(64 100)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 120)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 149)(82 150)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)

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

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

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

120 conjugacy classes

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

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 D9 C5×S3 D18 S3×C10 C5×D9 C10×D9 kernel D9×C2×C10 C10×D9 C2×C90 C22×D9 D18 C2×C18 C2×C30 C30 C2×C10 C2×C6 C10 C6 C22 C2 # reps 1 6 1 4 24 4 1 3 3 4 9 12 12 36

Matrix representation of D9×C2×C10 in GL4(𝔽181) generated by

 1 0 0 0 0 180 0 0 0 0 1 0 0 0 0 1
,
 180 0 0 0 0 1 0 0 0 0 59 0 0 0 0 59
,
 1 0 0 0 0 1 0 0 0 0 131 54 0 0 127 4
,
 1 0 0 0 0 1 0 0 0 0 4 50 0 0 54 177
G:=sub<GL(4,GF(181))| [1,0,0,0,0,180,0,0,0,0,1,0,0,0,0,1],[180,0,0,0,0,1,0,0,0,0,59,0,0,0,0,59],[1,0,0,0,0,1,0,0,0,0,131,127,0,0,54,4],[1,0,0,0,0,1,0,0,0,0,4,54,0,0,50,177] >;

D9×C2×C10 in GAP, Magma, Sage, TeX

D_9\times C_2\times C_{10}
% in TeX

G:=Group("D9xC2xC10");
// GroupNames label

G:=SmallGroup(360,48);
// by ID

G=gap.SmallGroup(360,48);
# by ID

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

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

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