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## G = C6×C12⋊S3order 432 = 24·33

### Direct product of C6 and C12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×C12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C2×C6×C3⋊S3 — C6×C12⋊S3
 Lower central C32 — C3×C6 — C6×C12⋊S3
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×C12⋊S3
G = < a,b,c,d | a6=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1412 in 388 conjugacy classes, 118 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×D12, C6×D4, C3×C3⋊S3, C32×C6, C32×C6, C3×D12, C12⋊S3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×C12, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C6×D12, C2×C12⋊S3, C3×C12⋊S3, C3×C6×C12, C2×C6×C3⋊S3, C6×C12⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, D12, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, C2×D12, C6×D4, C3×C3⋊S3, C3×D12, C12⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C6×D12, C2×C12⋊S3, C3×C12⋊S3, C2×C6×C3⋊S3, C6×C12⋊S3

Smallest permutation representation of C6×C12⋊S3
On 144 points
Generators in S144
(1 134 26 37 100 72)(2 135 27 38 101 61)(3 136 28 39 102 62)(4 137 29 40 103 63)(5 138 30 41 104 64)(6 139 31 42 105 65)(7 140 32 43 106 66)(8 141 33 44 107 67)(9 142 34 45 108 68)(10 143 35 46 97 69)(11 144 36 47 98 70)(12 133 25 48 99 71)(13 127 120 75 51 92)(14 128 109 76 52 93)(15 129 110 77 53 94)(16 130 111 78 54 95)(17 131 112 79 55 96)(18 132 113 80 56 85)(19 121 114 81 57 86)(20 122 115 82 58 87)(21 123 116 83 59 88)(22 124 117 84 60 89)(23 125 118 73 49 90)(24 126 119 74 50 91)
(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 34 104)(2 35 105)(3 36 106)(4 25 107)(5 26 108)(6 27 97)(7 28 98)(8 29 99)(9 30 100)(10 31 101)(11 32 102)(12 33 103)(13 59 112)(14 60 113)(15 49 114)(16 50 115)(17 51 116)(18 52 117)(19 53 118)(20 54 119)(21 55 120)(22 56 109)(23 57 110)(24 58 111)(37 68 138)(38 69 139)(39 70 140)(40 71 141)(41 72 142)(42 61 143)(43 62 144)(44 63 133)(45 64 134)(46 65 135)(47 66 136)(48 67 137)(73 121 94)(74 122 95)(75 123 96)(76 124 85)(77 125 86)(78 126 87)(79 127 88)(80 128 89)(81 129 90)(82 130 91)(83 131 92)(84 132 93)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 96)(12 95)(13 136)(14 135)(15 134)(16 133)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 132)(36 131)(37 110)(38 109)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 64)(50 63)(51 62)(52 61)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 98)(80 97)(81 108)(82 107)(83 106)(84 105)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3N 4A 4B 6A ··· 6F 6G ··· 6AP 6AQ ··· 6AX 12A ··· 12AZ order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 18 18 1 1 2 ··· 2 2 2 1 ··· 1 2 ··· 2 18 ··· 18 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 D12 C3×D4 S3×C6 S3×C6 C3×D12 kernel C6×C12⋊S3 C3×C12⋊S3 C3×C6×C12 C2×C6×C3⋊S3 C2×C12⋊S3 C12⋊S3 C6×C12 C22×C3⋊S3 C6×C12 C32×C6 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 # reps 1 4 1 2 2 8 2 4 4 2 8 4 8 16 4 16 8 32

Matrix representation of C6×C12⋊S3 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 9 8 0 0 0 0 0 3 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 4 0 0 0 0 7 5 0 0 0 0 0 0 6 3 0 0 0 0 10 7 0 0 0 0 0 0 6 2 0 0 0 0 2 7

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[9,0,0,0,0,0,8,3,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,7,0,0,0,0,4,5,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,6,2,0,0,0,0,2,7] >;

C6×C12⋊S3 in GAP, Magma, Sage, TeX

C_6\times C_{12}\rtimes S_3
% in TeX

G:=Group("C6xC12:S3");
// GroupNames label

G:=SmallGroup(432,712);
// by ID

G=gap.SmallGroup(432,712);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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