direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×D12, C21⋊6D4, C84⋊5C2, C28⋊3S3, C12⋊1C14, D6⋊1C14, C14.15D6, C42.20C22, C4⋊(S3×C7), C3⋊1(C7×D4), (S3×C14)⋊4C2, C2.4(S3×C14), C6.3(C2×C14), SmallGroup(168,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D12
G = < a,b,c | a7=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D12
►On 84 points
Generators in S84
(1 25 69 46 75 60 14)(2 26 70 47 76 49 15)(3 27 71 48 77 50 16)(4 28 72 37 78 51 17)(5 29 61 38 79 52 18)(6 30 62 39 80 53 19)(7 31 63 40 81 54 20)(8 32 64 41 82 55 21)(9 33 65 42 83 56 22)(10 34 66 43 84 57 23)(11 35 67 44 73 58 24)(12 36 68 45 74 59 13)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(73 76)(74 75)(77 84)(78 83)(79 82)(80 81)
G:=sub<Sym(84)| (1,25,69,46,75,60,14)(2,26,70,47,76,49,15)(3,27,71,48,77,50,16)(4,28,72,37,78,51,17)(5,29,61,38,79,52,18)(6,30,62,39,80,53,19)(7,31,63,40,81,54,20)(8,32,64,41,82,55,21)(9,33,65,42,83,56,22)(10,34,66,43,84,57,23)(11,35,67,44,73,58,24)(12,36,68,45,74,59,13), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81)>;
G:=Group( (1,25,69,46,75,60,14)(2,26,70,47,76,49,15)(3,27,71,48,77,50,16)(4,28,72,37,78,51,17)(5,29,61,38,79,52,18)(6,30,62,39,80,53,19)(7,31,63,40,81,54,20)(8,32,64,41,82,55,21)(9,33,65,42,83,56,22)(10,34,66,43,84,57,23)(11,35,67,44,73,58,24)(12,36,68,45,74,59,13), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81) );
G=PermutationGroup([[(1,25,69,46,75,60,14),(2,26,70,47,76,49,15),(3,27,71,48,77,50,16),(4,28,72,37,78,51,17),(5,29,61,38,79,52,18),(6,30,62,39,80,53,19),(7,31,63,40,81,54,20),(8,32,64,41,82,55,21),(9,33,65,42,83,56,22),(10,34,66,43,84,57,23),(11,35,67,44,73,58,24),(12,36,68,45,74,59,13)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,42),(38,41),(39,40),(43,48),(44,47),(45,46),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(73,76),(74,75),(77,84),(78,83),(79,82),(80,81)]])
C7×D12 is a maximal subgroup of
C21⋊D8 C7⋊D24 C42.D4 D12.D7 D12⋊D7 D12⋊5D7 C28⋊D6 S3×C7×D4
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 7A | ··· | 7F | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 7 | ··· | 7 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | S3 | D4 | D6 | D12 | S3×C7 | C7×D4 | S3×C14 | C7×D12 |
| kernel | C7×D12 | C84 | S3×C14 | D12 | C12 | D6 | C28 | C21 | C14 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 1 | 1 | 2 | 6 | 6 | 6 | 12 |
Matrix representation of C7×D12 ►in GL4(𝔽337) generated by
| 79 | 0 | 0 | 0 |
| 0 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 336 | 335 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 336 |
| 336 | 335 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 336 |
| 0 | 0 | 0 | 336 |
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,1,0,0,0,0,1],[336,1,0,0,335,1,0,0,0,0,0,1,0,0,336,336],[336,0,0,0,335,1,0,0,0,0,1,0,0,0,336,336] >;
C7×D12 in GAP, Magma, Sage, TeX
C_7\times D_{12} % in TeX
G:=Group("C7xD12"); // GroupNames label
G:=SmallGroup(168,31);
// by ID
G=gap.SmallGroup(168,31);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-3,301,146,2804]);
// Polycyclic
G:=Group<a,b,c|a^7=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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