direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×D7, C84⋊4C2, C28⋊6C6, Dic7⋊5C6, D14.2C6, C6.14D14, C42.14C22, C21⋊5(C2×C4), C7⋊4(C2×C12), C2.1(C6×D7), (C6×D7).2C2, C14.10(C2×C6), (C3×Dic7)⋊5C2, SmallGroup(168,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C12×D7 |
Generators and relations for C12×D7
G = < a,b,c | a12=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C12×D7
►On 84 points
Generators in S84
(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)
(1 47 82 24 52 71 31)(2 48 83 13 53 72 32)(3 37 84 14 54 61 33)(4 38 73 15 55 62 34)(5 39 74 16 56 63 35)(6 40 75 17 57 64 36)(7 41 76 18 58 65 25)(8 42 77 19 59 66 26)(9 43 78 20 60 67 27)(10 44 79 21 49 68 28)(11 45 80 22 50 69 29)(12 46 81 23 51 70 30)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
G:=sub<Sym(84)| (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), (1,47,82,24,52,71,31)(2,48,83,13,53,72,32)(3,37,84,14,54,61,33)(4,38,73,15,55,62,34)(5,39,74,16,56,63,35)(6,40,75,17,57,64,36)(7,41,76,18,58,65,25)(8,42,77,19,59,66,26)(9,43,78,20,60,67,27)(10,44,79,21,49,68,28)(11,45,80,22,50,69,29)(12,46,81,23,51,70,30), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)>;
G:=Group( (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), (1,47,82,24,52,71,31)(2,48,83,13,53,72,32)(3,37,84,14,54,61,33)(4,38,73,15,55,62,34)(5,39,74,16,56,63,35)(6,40,75,17,57,64,36)(7,41,76,18,58,65,25)(8,42,77,19,59,66,26)(9,43,78,20,60,67,27)(10,44,79,21,49,68,28)(11,45,80,22,50,69,29)(12,46,81,23,51,70,30), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78) );
G=PermutationGroup([[(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)], [(1,47,82,24,52,71,31),(2,48,83,13,53,72,32),(3,37,84,14,54,61,33),(4,38,73,15,55,62,34),(5,39,74,16,56,63,35),(6,40,75,17,57,64,36),(7,41,76,18,58,65,25),(8,42,77,19,59,66,26),(9,43,78,20,60,67,27),(10,44,79,21,49,68,28),(11,45,80,22,50,69,29),(12,46,81,23,51,70,30)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)]])
C12×D7 is a maximal subgroup of
C28.32D6 D6.D14 D12⋊5D7 D14.D6
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D7 | D14 | C3×D7 | C4×D7 | C6×D7 | C12×D7 |
| kernel | C12×D7 | C3×Dic7 | C84 | C6×D7 | C4×D7 | C3×D7 | Dic7 | C28 | D14 | D7 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 3 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C12×D7 ►in GL2(𝔽13) generated by
| 7 | 0 |
| 0 | 7 |
| 5 | 7 |
| 5 | 2 |
| 11 | 2 |
| 5 | 2 |
G:=sub<GL(2,GF(13))| [7,0,0,7],[5,5,7,2],[11,5,2,2] >;
C12×D7 in GAP, Magma, Sage, TeX
C_{12}\times D_7 % in TeX
G:=Group("C12xD7"); // GroupNames label
G:=SmallGroup(168,25);
// by ID
G=gap.SmallGroup(168,25);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,66,3604]);
// Polycyclic
G:=Group<a,b,c|a^12=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export