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## G = Dic3×C2×C18order 432 = 24·33

### Direct product of C2×C18 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C18
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C9×Dic3 — Dic3×C18 — Dic3×C2×C18
 Lower central C3 — Dic3×C2×C18
 Upper central C1 — C22×C18

Generators and relations for Dic3×C2×C18
G = < a,b,c,d | a2=b18=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 292 in 194 conjugacy classes, 129 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C3×C18, C3×C18, C2×C36, C22×C18, C22×C18, C6×Dic3, C2×C62, C9×Dic3, C6×C18, C22×C36, Dic3×C2×C6, Dic3×C18, C2×C6×C18, Dic3×C2×C18
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, C18, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C36, C2×C18, C3×Dic3, S3×C6, C22×Dic3, C22×C12, S3×C9, C2×C36, C22×C18, C6×Dic3, S3×C2×C6, C9×Dic3, S3×C18, C22×C36, Dic3×C2×C6, Dic3×C18, S3×C2×C18, Dic3×C2×C18

Smallest permutation representation of Dic3×C2×C18
On 144 points
Generators in S144
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 109)(15 110)(16 111)(17 112)(18 113)(19 50)(20 51)(21 52)(22 53)(23 54)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(55 144)(56 127)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 141)(71 142)(72 143)(73 107)(74 108)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)
(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)
(1 102 7 108 13 96)(2 103 8 91 14 97)(3 104 9 92 15 98)(4 105 10 93 16 99)(5 106 11 94 17 100)(6 107 12 95 18 101)(19 63 31 57 25 69)(20 64 32 58 26 70)(21 65 33 59 27 71)(22 66 34 60 28 72)(23 67 35 61 29 55)(24 68 36 62 30 56)(37 139 49 133 43 127)(38 140 50 134 44 128)(39 141 51 135 45 129)(40 142 52 136 46 130)(41 143 53 137 47 131)(42 144 54 138 48 132)(73 125 79 113 85 119)(74 126 80 114 86 120)(75 109 81 115 87 121)(76 110 82 116 88 122)(77 111 83 117 89 123)(78 112 84 118 90 124)
(1 56 108 36)(2 57 91 19)(3 58 92 20)(4 59 93 21)(5 60 94 22)(6 61 95 23)(7 62 96 24)(8 63 97 25)(9 64 98 26)(10 65 99 27)(11 66 100 28)(12 67 101 29)(13 68 102 30)(14 69 103 31)(15 70 104 32)(16 71 105 33)(17 72 106 34)(18 55 107 35)(37 120 133 80)(38 121 134 81)(39 122 135 82)(40 123 136 83)(41 124 137 84)(42 125 138 85)(43 126 139 86)(44 109 140 87)(45 110 141 88)(46 111 142 89)(47 112 143 90)(48 113 144 73)(49 114 127 74)(50 115 128 75)(51 116 129 76)(52 117 130 77)(53 118 131 78)(54 119 132 79)

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

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

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

216 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A ··· 4H 6A ··· 6N 6O ··· 6AI 9A ··· 9F 9G ··· 9L 12A ··· 12P 18A ··· 18AP 18AQ ··· 18CF 36A ··· 36AV order 1 2 ··· 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 ··· 1 1 1 2 2 2 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 S3×C9 C9×Dic3 S3×C18 kernel Dic3×C2×C18 Dic3×C18 C2×C6×C18 Dic3×C2×C6 C6×C18 C6×Dic3 C2×C62 C22×Dic3 C62 C2×Dic3 C22×C6 C2×C6 C22×C18 C2×C18 C2×C18 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 8 12 2 6 16 36 6 48 1 4 3 2 8 6 6 24 18

Matrix representation of Dic3×C2×C18 in GL4(𝔽37) generated by

 1 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 21 0 0 0 0 9 0 0 0 0 12 0 0 0 0 12
,
 36 0 0 0 0 36 0 0 0 0 11 0 0 0 0 27
,
 6 0 0 0 0 6 0 0 0 0 0 1 0 0 36 0
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,0,9,0,0,0,0,12,0,0,0,0,12],[36,0,0,0,0,36,0,0,0,0,11,0,0,0,0,27],[6,0,0,0,0,6,0,0,0,0,0,36,0,0,1,0] >;

Dic3×C2×C18 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{18}
% in TeX

G:=Group("Dic3xC2xC18");
// GroupNames label

G:=SmallGroup(432,373);
// by ID

G=gap.SmallGroup(432,373);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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