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## G = C6×D36order 432 = 24·33

### Direct product of C6 and D36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C6×D36
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C2×C6×D9 — C6×D36
 Lower central C9 — C18 — C6×D36
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×D36
G = < a,b,c | a6=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 782 in 178 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C3×C9, C36, C36, D18, D18, C2×C18, C2×C18, C3×C12, S3×C6, C62, C2×D12, C6×D4, C3×D9, C3×C18, C3×C18, D36, C2×C36, C2×C36, C22×D9, C3×D12, C6×C12, S3×C2×C6, C3×C36, C6×D9, C6×D9, C6×C18, C2×D36, C6×D12, C3×D36, C6×C36, C2×C6×D9, C6×D36
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, D9, C3×S3, D12, C3×D4, C22×S3, C22×C6, D18, S3×C6, C2×D12, C6×D4, C3×D9, D36, C22×D9, C3×D12, S3×C2×C6, C6×D9, C2×D36, C6×D12, C3×D36, C2×C6×D9, C6×D36

Smallest permutation representation of C6×D36
On 144 points
Generators in S144
(1 63 25 51 13 39)(2 64 26 52 14 40)(3 65 27 53 15 41)(4 66 28 54 16 42)(5 67 29 55 17 43)(6 68 30 56 18 44)(7 69 31 57 19 45)(8 70 32 58 20 46)(9 71 33 59 21 47)(10 72 34 60 22 48)(11 37 35 61 23 49)(12 38 36 62 24 50)(73 110 85 122 97 134)(74 111 86 123 98 135)(75 112 87 124 99 136)(76 113 88 125 100 137)(77 114 89 126 101 138)(78 115 90 127 102 139)(79 116 91 128 103 140)(80 117 92 129 104 141)(81 118 93 130 105 142)(82 119 94 131 106 143)(83 120 95 132 107 144)(84 121 96 133 108 109)
(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 110)(9 109)(10 144)(11 143)(12 142)(13 141)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 120)(35 119)(36 118)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)

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

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

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

126 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 ··· 9I 12A ··· 12P 18A ··· 18AA 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 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 18 18 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 18 ··· 18 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 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 type + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 D9 C3×S3 C3×D4 D12 D18 S3×C6 D18 S3×C6 C3×D9 D36 C3×D12 C6×D9 C6×D9 C3×D36 kernel C6×D36 C3×D36 C6×C36 C2×C6×D9 C2×D36 D36 C2×C36 C22×D9 C6×C12 C3×C18 C3×C12 C62 C2×C12 C2×C12 C18 C3×C6 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 1 2 2 1 3 2 4 4 6 4 3 2 6 12 8 12 6 24

Matrix representation of C6×D36 in GL3(𝔽37) generated by

 11 0 0 0 27 0 0 0 27
,
 1 0 0 0 32 0 0 0 22
,
 36 0 0 0 0 21 0 30 0
G:=sub<GL(3,GF(37))| [11,0,0,0,27,0,0,0,27],[1,0,0,0,32,0,0,0,22],[36,0,0,0,0,30,0,21,0] >;

C6×D36 in GAP, Magma, Sage, TeX

C_6\times D_{36}
% in TeX

G:=Group("C6xD36");
// GroupNames label

G:=SmallGroup(432,343);
// by ID

G=gap.SmallGroup(432,343);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^6=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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