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## G = C3×D18⋊C4order 432 = 24·33

### Direct product of C3 and D18⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×D18⋊C4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C6×C18 — C2×C6×D9 — C3×D18⋊C4
 Lower central C9 — C18 — C3×D18⋊C4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×D18⋊C4
G = < a,b,c,d | a3=b18=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b9c >

Subgroups: 470 in 118 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, D6⋊C4, C3×C22⋊C4, C3×D9, C3×C18, C2×Dic9, C2×C36, C2×C36, C22×D9, C6×Dic3, C6×C12, S3×C2×C6, C3×Dic9, C3×C36, C6×D9, C6×D9, C6×C18, D18⋊C4, C3×D6⋊C4, C6×Dic9, C6×C36, C2×C6×D9, C3×D18⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, D9, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, D18, S3×C6, D6⋊C4, C3×C22⋊C4, C3×D9, C4×D9, D36, C9⋊D4, S3×C12, C3×D12, C3×C3⋊D4, C6×D9, D18⋊C4, C3×D6⋊C4, C12×D9, C3×D36, C3×C9⋊D4, C3×D18⋊C4

Smallest permutation representation of C3×D18⋊C4
On 144 points
Generators in S144
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)(109 121 115)(110 122 116)(111 123 117)(112 124 118)(113 125 119)(114 126 120)(127 139 133)(128 140 134)(129 141 135)(130 142 136)(131 143 137)(132 144 138)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 92)(20 91)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 113)(38 112)(39 111)(40 110)(41 109)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(61 129)(62 128)(63 127)(64 144)(65 143)(66 142)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)
(1 55 31 38)(2 56 32 39)(3 57 33 40)(4 58 34 41)(5 59 35 42)(6 60 36 43)(7 61 19 44)(8 62 20 45)(9 63 21 46)(10 64 22 47)(11 65 23 48)(12 66 24 49)(13 67 25 50)(14 68 26 51)(15 69 27 52)(16 70 28 53)(17 71 29 54)(18 72 30 37)(73 137 91 114)(74 138 92 115)(75 139 93 116)(76 140 94 117)(77 141 95 118)(78 142 96 119)(79 143 97 120)(80 144 98 121)(81 127 99 122)(82 128 100 123)(83 129 101 124)(84 130 102 125)(85 131 103 126)(86 132 104 109)(87 133 105 110)(88 134 106 111)(89 135 107 112)(90 136 108 113)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 9A ··· 9I 12A ··· 12P 12Q 12R 12S 12T 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 1 1 2 2 2 2 2 18 18 1 ··· 1 2 ··· 2 18 18 18 18 2 ··· 2 2 ··· 2 18 18 18 18 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 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 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D6 D9 C3×S3 C3×D4 C4×S3 D12 C3⋊D4 D18 S3×C6 C3×D9 C4×D9 D36 C9⋊D4 S3×C12 C3×D12 C3×C3⋊D4 C6×D9 C12×D9 C3×D36 C3×C9⋊D4 kernel C3×D18⋊C4 C6×Dic9 C6×C36 C2×C6×D9 D18⋊C4 C6×D9 C2×Dic9 C2×C36 C22×D9 D18 C6×C12 C3×C18 C62 C2×C12 C2×C12 C18 C3×C6 C3×C6 C3×C6 C2×C6 C2×C6 C2×C4 C6 C6 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 3 2 4 2 2 2 3 2 6 6 6 6 4 4 4 6 12 12 12

Matrix representation of C3×D18⋊C4 in GL3(𝔽37) generated by

 26 0 0 0 26 0 0 0 26
,
 1 0 0 0 30 0 0 0 21
,
 36 0 0 0 0 21 0 30 0
,
 6 0 0 0 36 0 0 0 1
G:=sub<GL(3,GF(37))| [26,0,0,0,26,0,0,0,26],[1,0,0,0,30,0,0,0,21],[36,0,0,0,0,30,0,21,0],[6,0,0,0,36,0,0,0,1] >;

C3×D18⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_{18}\rtimes C_4
% in TeX

G:=Group("C3xD18:C4");
// GroupNames label

G:=SmallGroup(432,134);
// by ID

G=gap.SmallGroup(432,134);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^9*c>;
// generators/relations

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