direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C6×C12, C12⋊3C62, D6.4C62, Dic3⋊3C62, C62.168D6, C6⋊1(C6×C12), (C6×C12)⋊16C6, C2.1(S3×C62), C6.2(C2×C62), (C6×Dic3)⋊11C6, (S3×C62).6C2, (C2×C6).20C62, C62.71(C2×C6), C33⋊16(C22×C4), C32⋊8(C22×C12), (C32×C12)⋊16C22, (C32×C6).76C23, (C3×C62).57C22, (C32×Dic3)⋊24C22, C3⋊1(C2×C6×C12), (C3×C6×C12)⋊11C2, C6.74(S3×C2×C6), (C3×C6)⋊7(C2×C12), (C2×C12)⋊5(C3×C6), (S3×C2×C6).11C6, (C3×C12)⋊13(C2×C6), C22.9(S3×C3×C6), (C2×C6).96(S3×C6), (Dic3×C3×C6)⋊17C2, (S3×C6).22(C2×C6), (C32×C6)⋊10(C2×C4), (C2×Dic3)⋊5(C3×C6), (S3×C3×C6).33C22, (C3×Dic3)⋊10(C2×C6), (C22×S3).2(C3×C6), (C3×C6).50(C22×C6), (C3×C6).195(C22×S3), SmallGroup(432,701)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C6×C12 |
Generators and relations for S3×C6×C12
G = < a,b,c,d | a6=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 680 in 388 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C62, C62, S3×C2×C4, C22×C12, S3×C32, C32×C6, C32×C6, S3×C12, C6×Dic3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C2×C62, C32×Dic3, C32×C12, S3×C3×C6, C3×C62, S3×C2×C12, C2×C6×C12, S3×C3×C12, Dic3×C3×C6, C3×C6×C12, S3×C62, S3×C6×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C32, C12, D6, C2×C6, C22×C4, C3×S3, C3×C6, C4×S3, C2×C12, C22×S3, C22×C6, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, S3×C32, S3×C12, C6×C12, S3×C2×C6, C2×C62, S3×C3×C6, S3×C2×C12, C2×C6×C12, S3×C3×C12, S3×C62, S3×C6×C12
►On 144 points
Generators in S144
(1 114 143 50 77 86)(2 115 144 51 78 87)(3 116 133 52 79 88)(4 117 134 53 80 89)(5 118 135 54 81 90)(6 119 136 55 82 91)(7 120 137 56 83 92)(8 109 138 57 84 93)(9 110 139 58 73 94)(10 111 140 59 74 95)(11 112 141 60 75 96)(12 113 142 49 76 85)(13 36 42 124 97 70)(14 25 43 125 98 71)(15 26 44 126 99 72)(16 27 45 127 100 61)(17 28 46 128 101 62)(18 29 47 129 102 63)(19 30 48 130 103 64)(20 31 37 131 104 65)(21 32 38 132 105 66)(22 33 39 121 106 67)(23 34 40 122 107 68)(24 35 41 123 108 69)
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)(97 105 101)(98 106 102)(99 107 103)(100 108 104)(109 113 117)(110 114 118)(111 115 119)(112 116 120)(121 129 125)(122 130 126)(123 131 127)(124 132 128)(133 137 141)(134 138 142)(135 139 143)(136 140 144)
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 116)(26 117)(27 118)(28 119)(29 120)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 121)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)
G:=sub<Sym(144)| (1,114,143,50,77,86)(2,115,144,51,78,87)(3,116,133,52,79,88)(4,117,134,53,80,89)(5,118,135,54,81,90)(6,119,136,55,82,91)(7,120,137,56,83,92)(8,109,138,57,84,93)(9,110,139,58,73,94)(10,111,140,59,74,95)(11,112,141,60,75,96)(12,113,142,49,76,85)(13,36,42,124,97,70)(14,25,43,125,98,71)(15,26,44,126,99,72)(16,27,45,127,100,61)(17,28,46,128,101,62)(18,29,47,129,102,63)(19,30,48,130,103,64)(20,31,37,131,104,65)(21,32,38,132,105,66)(22,33,39,121,106,67)(23,34,40,122,107,68)(24,35,41,123,108,69), (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,113,117)(110,114,118)(111,115,119)(112,116,120)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,116)(26,117)(27,118)(28,119)(29,120)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)>;
G:=Group( (1,114,143,50,77,86)(2,115,144,51,78,87)(3,116,133,52,79,88)(4,117,134,53,80,89)(5,118,135,54,81,90)(6,119,136,55,82,91)(7,120,137,56,83,92)(8,109,138,57,84,93)(9,110,139,58,73,94)(10,111,140,59,74,95)(11,112,141,60,75,96)(12,113,142,49,76,85)(13,36,42,124,97,70)(14,25,43,125,98,71)(15,26,44,126,99,72)(16,27,45,127,100,61)(17,28,46,128,101,62)(18,29,47,129,102,63)(19,30,48,130,103,64)(20,31,37,131,104,65)(21,32,38,132,105,66)(22,33,39,121,106,67)(23,34,40,122,107,68)(24,35,41,123,108,69), (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,113,117)(110,114,118)(111,115,119)(112,116,120)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,116)(26,117)(27,118)(28,119)(29,120)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103) );
G=PermutationGroup([[(1,114,143,50,77,86),(2,115,144,51,78,87),(3,116,133,52,79,88),(4,117,134,53,80,89),(5,118,135,54,81,90),(6,119,136,55,82,91),(7,120,137,56,83,92),(8,109,138,57,84,93),(9,110,139,58,73,94),(10,111,140,59,74,95),(11,112,141,60,75,96),(12,113,142,49,76,85),(13,36,42,124,97,70),(14,25,43,125,98,71),(15,26,44,126,99,72),(16,27,45,127,100,61),(17,28,46,128,101,62),(18,29,47,129,102,63),(19,30,48,130,103,64),(20,31,37,131,104,65),(21,32,38,132,105,66),(22,33,39,121,106,67),(23,34,40,122,107,68),(24,35,41,123,108,69)], [(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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,89,93),(86,90,94),(87,91,95),(88,92,96),(97,105,101),(98,106,102),(99,107,103),(100,108,104),(109,113,117),(110,114,118),(111,115,119),(112,116,120),(121,129,125),(122,130,126),(123,131,127),(124,132,128),(133,137,141),(134,138,142),(135,139,143),(136,140,144)], [(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(25,116),(26,117),(27,118),(28,119),(29,120),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,121),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,85),(69,86),(70,87),(71,88),(72,89),(73,104),(74,105),(75,106),(76,107),(77,108),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6X | 6Y | ··· | 6AY | 6AZ | ··· | 6CE | 12A | ··· | 12AF | 12AG | ··· | 12BP | 12BQ | ··· | 12CV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C6 | S3×C12 |
| kernel | S3×C6×C12 | S3×C3×C12 | Dic3×C3×C6 | C3×C6×C12 | S3×C62 | S3×C2×C12 | S3×C3×C6 | S3×C12 | C6×Dic3 | C6×C12 | S3×C2×C6 | S3×C6 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 32 | 8 | 8 | 8 | 64 | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 |
Matrix representation of S3×C6×C12 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 10 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 4 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,4,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,3,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,4,3],[1,0,0,0,0,12,0,0,0,0,12,8,0,0,0,1] >;
S3×C6×C12 in GAP, Magma, Sage, TeX
S_3\times C_6\times C_{12} % in TeX
G:=Group("S3xC6xC12"); // GroupNames label
G:=SmallGroup(432,701);
// by ID
G=gap.SmallGroup(432,701);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,394,14118]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations