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## G = C18×He3order 486 = 2·35

### Direct product of C18 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C18×He3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C9×He3 — C18×He3
 Lower central C1 — C3 — C18×He3
 Upper central C1 — C3×C18 — C18×He3

Generators and relations for C18×He3
G = < a,b,c,d | a18=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 378 in 202 conjugacy classes, 114 normal (14 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C32, C32, C32, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, He3, C33, C3×C18, C3×C18, C3×C18, C2×He3, C32×C6, C32⋊C9, C32×C9, C3×He3, C2×C32⋊C9, C32×C18, C6×He3, C9×He3, C18×He3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, C33, C3×C18, C2×He3, C32×C6, C32×C9, C3×He3, C9○He3, C32×C18, C6×He3, C2×C9○He3, C9×He3, C18×He3

Smallest permutation representation of C18×He3
On 162 points
Generators in S162
(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)
(1 40 100)(2 41 101)(3 42 102)(4 43 103)(5 44 104)(6 45 105)(7 46 106)(8 47 107)(9 48 108)(10 49 91)(11 50 92)(12 51 93)(13 52 94)(14 53 95)(15 54 96)(16 37 97)(17 38 98)(18 39 99)(19 86 138)(20 87 139)(21 88 140)(22 89 141)(23 90 142)(24 73 143)(25 74 144)(26 75 127)(27 76 128)(28 77 129)(29 78 130)(30 79 131)(31 80 132)(32 81 133)(33 82 134)(34 83 135)(35 84 136)(36 85 137)(55 115 158)(56 116 159)(57 117 160)(58 118 161)(59 119 162)(60 120 145)(61 121 146)(62 122 147)(63 123 148)(64 124 149)(65 125 150)(66 126 151)(67 109 152)(68 110 153)(69 111 154)(70 112 155)(71 113 156)(72 114 157)
(1 88 69)(2 89 70)(3 90 71)(4 73 72)(5 74 55)(6 75 56)(7 76 57)(8 77 58)(9 78 59)(10 79 60)(11 80 61)(12 81 62)(13 82 63)(14 83 64)(15 84 65)(16 85 66)(17 86 67)(18 87 68)(19 152 98)(20 153 99)(21 154 100)(22 155 101)(23 156 102)(24 157 103)(25 158 104)(26 159 105)(27 160 106)(28 161 107)(29 162 108)(30 145 91)(31 146 92)(32 147 93)(33 148 94)(34 149 95)(35 150 96)(36 151 97)(37 137 126)(38 138 109)(39 139 110)(40 140 111)(41 141 112)(42 142 113)(43 143 114)(44 144 115)(45 127 116)(46 128 117)(47 129 118)(48 130 119)(49 131 120)(50 132 121)(51 133 122)(52 134 123)(53 135 124)(54 136 125)
(1 148 46)(2 149 47)(3 150 48)(4 151 49)(5 152 50)(6 153 51)(7 154 52)(8 155 53)(9 156 54)(10 157 37)(11 158 38)(12 159 39)(13 160 40)(14 161 41)(15 162 42)(16 145 43)(17 146 44)(18 147 45)(19 121 55)(20 122 56)(21 123 57)(22 124 58)(23 125 59)(24 126 60)(25 109 61)(26 110 62)(27 111 63)(28 112 64)(29 113 65)(30 114 66)(31 115 67)(32 116 68)(33 117 69)(34 118 70)(35 119 71)(36 120 72)(73 97 131)(74 98 132)(75 99 133)(76 100 134)(77 101 135)(78 102 136)(79 103 137)(80 104 138)(81 105 139)(82 106 140)(83 107 141)(84 108 142)(85 91 143)(86 92 144)(87 93 127)(88 94 128)(89 95 129)(90 96 130)

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

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

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

198 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 6A ··· 6H 6I ··· 6AF 9A ··· 9R 9S ··· 9BN 18A ··· 18R 18S ··· 18BN order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 He3 C2×He3 C9○He3 C2×C9○He3 kernel C18×He3 C9×He3 C2×C32⋊C9 C32×C18 C6×He3 C32⋊C9 C32×C9 C3×He3 C2×He3 He3 C18 C9 C6 C3 # reps 1 1 16 8 2 16 8 2 54 54 6 6 12 12

Matrix representation of C18×He3 in GL4(𝔽19) generated by

 4 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 1 0 0 0 0 11 0 0 0 0 0 7 0 4 18 8
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 11 18 0 4 18 8 0 0 11 0
G:=sub<GL(4,GF(19))| [4,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,11,0,4,0,0,0,18,0,0,7,8],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,4,0,0,11,18,11,0,18,8,0] >;

C18×He3 in GAP, Magma, Sage, TeX

C_{18}\times {\rm He}_3
% in TeX

G:=Group("C18xHe3");
// GroupNames label

G:=SmallGroup(486,194);
// by ID

G=gap.SmallGroup(486,194);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,237]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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