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## G = C18×D12order 432 = 24·33

### Direct product of C18 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C18×D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C2×C18 — C18×D12
 Lower central C3 — C6 — C18×D12
 Upper central C1 — C2×C18 — C2×C36

Generators and relations for C18×D12
G = < a,b,c | a18=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 404 in 178 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, S3×C6, S3×C6, C62, C2×D12, C6×D4, S3×C9, C3×C18, C3×C18, C2×C36, C2×C36, D4×C9, C22×C18, C3×D12, C6×C12, S3×C2×C6, C3×C36, S3×C18, S3×C18, C6×C18, D4×C18, C6×D12, C9×D12, C6×C36, S3×C2×C18, C18×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C9, D6, C2×C6, C2×D4, C18, C3×S3, D12, C3×D4, C22×S3, C22×C6, C2×C18, S3×C6, C2×D12, C6×D4, S3×C9, D4×C9, C22×C18, C3×D12, S3×C2×C6, S3×C18, D4×C18, C6×D12, C9×D12, S3×C2×C18, C18×D12

Smallest permutation representation of C18×D12
On 144 points
Generators in S144
(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)
(1 75 113 140 7 81 119 128 13 87 125 134)(2 76 114 141 8 82 120 129 14 88 126 135)(3 77 115 142 9 83 121 130 15 89 109 136)(4 78 116 143 10 84 122 131 16 90 110 137)(5 79 117 144 11 85 123 132 17 73 111 138)(6 80 118 127 12 86 124 133 18 74 112 139)(19 61 50 98 31 55 44 92 25 67 38 104)(20 62 51 99 32 56 45 93 26 68 39 105)(21 63 52 100 33 57 46 94 27 69 40 106)(22 64 53 101 34 58 47 95 28 70 41 107)(23 65 54 102 35 59 48 96 29 71 42 108)(24 66 37 103 36 60 49 97 30 72 43 91)
(1 34)(2 35)(3 36)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 127)(70 128)(71 129)(72 130)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6O 6P ··· 6W 9A ··· 9F 9G ··· 9L 12A ··· 12P 18A ··· 18R 18S ··· 18AJ 18AK ··· 18BH 36A ··· 36AJ order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 6 6 6 6 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

162 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 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 D6 C3×S3 D12 C3×D4 S3×C6 S3×C6 S3×C9 D4×C9 C3×D12 S3×C18 S3×C18 C9×D12 kernel C18×D12 C9×D12 C6×C36 S3×C2×C18 C6×D12 C3×D12 C6×C12 S3×C2×C6 C2×D12 D12 C2×C12 C22×S3 C2×C36 C3×C18 C36 C2×C18 C2×C12 C18 C3×C6 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 6 24 6 12 1 2 2 1 2 4 4 4 2 6 12 8 12 6 24

Matrix representation of C18×D12 in GL4(𝔽37) generated by

 25 0 0 0 0 25 0 0 0 0 33 0 0 0 0 33
,
 10 0 0 0 0 26 0 0 0 0 8 0 0 0 22 14
,
 0 26 0 0 10 0 0 0 0 0 18 15 0 0 13 19
G:=sub<GL(4,GF(37))| [25,0,0,0,0,25,0,0,0,0,33,0,0,0,0,33],[10,0,0,0,0,26,0,0,0,0,8,22,0,0,0,14],[0,10,0,0,26,0,0,0,0,0,18,13,0,0,15,19] >;

C18×D12 in GAP, Magma, Sage, TeX

C_{18}\times D_{12}
% in TeX

G:=Group("C18xD12");
// GroupNames label

G:=SmallGroup(432,346);
// by ID

G=gap.SmallGroup(432,346);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^18=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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