direct product, metabelian, supersoluble, monomial
Aliases: C3×C6×D12, C12⋊2C62, D6⋊1C62, C62.169D6, C12⋊8(S3×C6), (C6×C12)⋊15C6, (C6×C12)⋊18S3, (C3×C12)⋊26D6, C6⋊1(D4×C32), C33⋊29(C2×D4), (S3×C62)⋊7C2, C32⋊12(C6×D4), (C32×C6)⋊10D4, C2.4(S3×C62), C6.3(C2×C62), (C2×C6).21C62, C62.72(C2×C6), (C32×C12)⋊14C22, (C32×C6).77C23, (C3×C62).58C22, C4⋊2(S3×C3×C6), C3⋊1(D4×C3×C6), (S3×C2×C6)⋊6C6, (C3×C6×C12)⋊9C2, (C3×C6)⋊7(C3×D4), C6.75(S3×C2×C6), (S3×C6)⋊7(C2×C6), (C2×C12)⋊3(C3×C6), (C2×C12)⋊6(C3×S3), (C3×C12)⋊12(C2×C6), (C2×C6).97(S3×C6), (S3×C3×C6)⋊21C22, (C2×C4)⋊2(S3×C32), C22.10(S3×C3×C6), (C22×S3)⋊2(C3×C6), (C3×C6).51(C22×C6), (C3×C6).196(C22×S3), SmallGroup(432,702)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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