direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×D7, C7≀C2, AΣL1(𝔽49), C7⋊C14, C72⋊1C2, SmallGroup(98,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×D7 |
Generators and relations for C7×D7
G = < a,b,c | a7=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C7×D7
►On 14 points - transitive group 14T8
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 7 6 5 4 3 2)(8 9 10 11 12 13 14)
(1 14)(2 8)(3 9)(4 10)(5 11)(6 12)(7 13)
G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,7,6,5,4,3,2)(8,9,10,11,12,13,14), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,7,6,5,4,3,2)(8,9,10,11,12,13,14), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,7,6,5,4,3,2),(8,9,10,11,12,13,14)], [(1,14),(2,8),(3,9),(4,10),(5,11),(6,12),(7,13)]])
G:=TransitiveGroup(14,8);
C7×D7 is a maximal subgroup of
C72⋊S3 C7⋊3F7 C7⋊4F7
| action | f(x) | Disc(f) |
|---|---|---|
| 14T8 | x14+28x11+28x10-28x9+140x8+360x7+147x6+196x5+336x4-546x3-532x2+896x+823 | -214·725·192·372·1272·2772·5212·200112 |
35 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F |
| order | 1 | 2 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D7 | C7×D7 |
| kernel | C7×D7 | C72 | D7 | C7 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 3 | 18 |
Matrix representation of C7×D7 ►in GL2(𝔽29) generated by
| 23 | 0 |
| 0 | 23 |
| 7 | 0 |
| 0 | 25 |
| 0 | 25 |
| 7 | 0 |
G:=sub<GL(2,GF(29))| [23,0,0,23],[7,0,0,25],[0,7,25,0] >;
C7×D7 in GAP, Magma, Sage, TeX
C_7\times D_7
% in TeX
G:=Group("C7xD7"); // GroupNames label
G:=SmallGroup(98,3);
// by ID
G=gap.SmallGroup(98,3);
# by ID
G:=PCGroup([3,-2,-7,-7,758]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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