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## G = D5×C2×C18order 360 = 23·32·5

### Direct product of C2×C18 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C18
 Chief series C1 — C5 — C15 — C45 — C9×D5 — D5×C18 — D5×C2×C18
 Lower central C5 — D5×C2×C18
 Upper central C1 — C2×C18

Generators and relations for D5×C2×C18
G = < a,b,c,d | a2=b18=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 228 in 96 conjugacy classes, 63 normal (15 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C6, C23, C9, D5, C10, C2×C6, C2×C6, C15, C18, C18, D10, C2×C10, C22×C6, C3×D5, C30, C2×C18, C2×C18, C22×D5, C45, C6×D5, C2×C30, C22×C18, C9×D5, C90, D5×C2×C6, D5×C18, C2×C90, D5×C2×C18
Quotients: C1, C2, C3, C22, C6, C23, C9, D5, C2×C6, C18, D10, C22×C6, C3×D5, C2×C18, C22×D5, C6×D5, C22×C18, C9×D5, D5×C2×C6, D5×C18, D5×C2×C18

Smallest permutation representation of D5×C2×C18
On 180 points
Generators in S180
(1 117)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 109)(12 110)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 88)(38 89)(39 90)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 140)(56 141)(57 142)(58 143)(59 144)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 177)(152 178)(153 179)(154 180)(155 163)(156 164)(157 165)(158 166)(159 167)(160 168)(161 169)(162 170)
(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)
(1 152 29 61 45)(2 153 30 62 46)(3 154 31 63 47)(4 155 32 64 48)(5 156 33 65 49)(6 157 34 66 50)(7 158 35 67 51)(8 159 36 68 52)(9 160 19 69 53)(10 161 20 70 54)(11 162 21 71 37)(12 145 22 72 38)(13 146 23 55 39)(14 147 24 56 40)(15 148 25 57 41)(16 149 26 58 42)(17 150 27 59 43)(18 151 28 60 44)(73 112 173 104 141)(74 113 174 105 142)(75 114 175 106 143)(76 115 176 107 144)(77 116 177 108 127)(78 117 178 91 128)(79 118 179 92 129)(80 119 180 93 130)(81 120 163 94 131)(82 121 164 95 132)(83 122 165 96 133)(84 123 166 97 134)(85 124 167 98 135)(86 125 168 99 136)(87 126 169 100 137)(88 109 170 101 138)(89 110 171 102 139)(90 111 172 103 140)
(1 54)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(61 161)(62 162)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(127 168)(128 169)(129 170)(130 171)(131 172)(132 173)(133 174)(134 175)(135 176)(136 177)(137 178)(138 179)(139 180)(140 163)(141 164)(142 165)(143 166)(144 167)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 5A 5B 6A ··· 6F 6G ··· 6N 9A ··· 9F 10A ··· 10F 15A 15B 15C 15D 18A ··· 18R 18S ··· 18AP 30A ··· 30L 45A ··· 45L 90A ··· 90AJ order 1 2 2 2 2 2 2 2 3 3 5 5 6 ··· 6 6 ··· 6 9 ··· 9 10 ··· 10 15 15 15 15 18 ··· 18 18 ··· 18 30 ··· 30 45 ··· 45 90 ··· 90 size 1 1 1 1 5 5 5 5 1 1 2 2 1 ··· 1 5 ··· 5 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 D5 D10 C3×D5 C6×D5 C9×D5 D5×C18 kernel D5×C2×C18 D5×C18 C2×C90 D5×C2×C6 C6×D5 C2×C30 C22×D5 D10 C2×C10 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 6 36 6 2 6 4 12 12 36

Matrix representation of D5×C2×C18 in GL4(𝔽181) generated by

 180 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 101 0 0 0 0 133 0 0 0 0 48 0 0 0 0 48
,
 1 0 0 0 0 1 0 0 0 0 168 1 0 0 12 180
,
 1 0 0 0 0 1 0 0 0 0 180 180 0 0 0 1
G:=sub<GL(4,GF(181))| [180,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[101,0,0,0,0,133,0,0,0,0,48,0,0,0,0,48],[1,0,0,0,0,1,0,0,0,0,168,12,0,0,1,180],[1,0,0,0,0,1,0,0,0,0,180,0,0,0,180,1] >;

D5×C2×C18 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{18}
% in TeX

G:=Group("D5xC2xC18");
// GroupNames label

G:=SmallGroup(360,47);
// by ID

G=gap.SmallGroup(360,47);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,93,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^5=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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