Copied to
clipboard

## G = C3×C6×D12order 432 = 24·33

### Direct product of C3×C6 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C6×D12
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C62 — C3×C6×D12
 Lower central C3 — C6 — C3×C6×D12
 Upper central C1 — C62 — C6×C12

Generators and relations for C3×C6×D12
G = < a,b,c,d | a3=b6=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 872 in 388 conjugacy classes, 162 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, D6, C2×C6, C2×C6, C2×C6, C2×D4, 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, S3×C6, C62, C62, C62, C2×D12, C6×D4, S3×C32, C32×C6, C32×C6, C3×D12, C6×C12, C6×C12, C6×C12, D4×C32, S3×C2×C6, C2×C62, C32×C12, S3×C3×C6, S3×C3×C6, C3×C62, C6×D12, D4×C3×C6, C32×D12, C3×C6×C12, S3×C62, C3×C6×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C32, D6, C2×C6, C2×D4, C3×S3, C3×C6, D12, C3×D4, C22×S3, C22×C6, S3×C6, C62, C2×D12, C6×D4, S3×C32, C3×D12, D4×C32, S3×C2×C6, C2×C62, S3×C3×C6, C6×D12, D4×C3×C6, C32×D12, S3×C62, C3×C6×D12

Smallest permutation representation of C3×C6×D12
On 144 points
Generators in S144
(1 39 31)(2 40 32)(3 41 33)(4 42 34)(5 43 35)(6 44 36)(7 45 25)(8 46 26)(9 47 27)(10 48 28)(11 37 29)(12 38 30)(13 129 138)(14 130 139)(15 131 140)(16 132 141)(17 121 142)(18 122 143)(19 123 144)(20 124 133)(21 125 134)(22 126 135)(23 127 136)(24 128 137)(49 70 76)(50 71 77)(51 72 78)(52 61 79)(53 62 80)(54 63 81)(55 64 82)(56 65 83)(57 66 84)(58 67 73)(59 68 74)(60 69 75)(85 106 116)(86 107 117)(87 108 118)(88 97 119)(89 98 120)(90 99 109)(91 100 110)(92 101 111)(93 102 112)(94 103 113)(95 104 114)(96 105 115)
(1 63 5 67 9 71)(2 64 6 68 10 72)(3 65 7 69 11 61)(4 66 8 70 12 62)(13 116 21 112 17 120)(14 117 22 113 18 109)(15 118 23 114 19 110)(16 119 24 115 20 111)(25 60 29 52 33 56)(26 49 30 53 34 57)(27 50 31 54 35 58)(28 51 32 55 36 59)(37 79 41 83 45 75)(38 80 42 84 46 76)(39 81 43 73 47 77)(40 82 44 74 48 78)(85 125 93 121 89 129)(86 126 94 122 90 130)(87 127 95 123 91 131)(88 128 96 124 92 132)(97 137 105 133 101 141)(98 138 106 134 102 142)(99 139 107 135 103 143)(100 140 108 136 104 144)
(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)
(1 100)(2 99)(3 98)(4 97)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 84)(25 85)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 112)(38 111)(39 110)(40 109)(41 120)(42 119)(43 118)(44 117)(45 116)(46 115)(47 114)(48 113)(49 124)(50 123)(51 122)(52 121)(53 132)(54 131)(55 130)(56 129)(57 128)(58 127)(59 126)(60 125)(61 142)(62 141)(63 140)(64 139)(65 138)(66 137)(67 136)(68 135)(69 134)(70 133)(71 144)(72 143)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6X 6Y ··· 6AY 6AZ ··· 6CE 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 6 6 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 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 C3×C6×D12 C32×D12 C3×C6×C12 S3×C62 C6×D12 C3×D12 C6×C12 S3×C2×C6 C6×C12 C32×C6 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 # reps 1 4 1 2 8 32 8 16 1 2 2 1 8 4 16 16 8 32

Matrix representation of C3×C6×D12 in GL4(𝔽13) generated by

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

C3×C6×D12 in GAP, Magma, Sage, TeX

C_3\times C_6\times D_{12}
% in TeX

G:=Group("C3xC6xD12");
// GroupNames label

G:=SmallGroup(432,702);
// by ID

G=gap.SmallGroup(432,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1598,394,14118]);
// Polycyclic

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

׿
×
𝔽