direct product, metabelian, supersoluble, monomial
Aliases: C6×C12⋊S3, C62.154D6, C12⋊6(S3×C6), (C3×C6)⋊7D12, C6⋊1(C3×D12), C3⋊2(C6×D12), (C6×C12)⋊15S3, (C6×C12)⋊12C6, (C3×C12)⋊23D6, C33⋊30(C2×D4), (C32×C6)⋊11D4, C32⋊13(C6×D4), C32⋊14(C2×D12), C62.77(C2×C6), (C32×C12)⋊15C22, (C3×C62).63C22, (C32×C6).87C23, C4⋊2(C6×C3⋊S3), C12⋊9(C2×C3⋊S3), (C3×C6×C12)⋊10C2, (C3×C6)⋊8(C3×D4), C6.54(S3×C2×C6), (C2×C12)⋊3(C3×S3), (C2×C12)⋊6(C3⋊S3), (C3×C12)⋊11(C2×C6), (C2×C6).77(S3×C6), (C22×C3⋊S3)⋊10C6, (C6×C3⋊S3)⋊21C22, C6.54(C22×C3⋊S3), C22.10(C6×C3⋊S3), (C3×C6).61(C22×C6), (C3×C6).176(C22×S3), (C2×C6×C3⋊S3)⋊8C2, C2.4(C2×C6×C3⋊S3), (C2×C4)⋊2(C3×C3⋊S3), (C2×C3⋊S3)⋊9(C2×C6), (C2×C6).69(C2×C3⋊S3), SmallGroup(432,712)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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