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## G = C7×D7order 98 = 2·72

### Direct product of C7 and D7

Aliases: C7×D7, C7C2, AΣL1(𝔽49), C7⋊C14, C721C2, SmallGroup(98,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7×D7
 Chief series C1 — C7 — C72 — C7×D7
 Lower central C7 — C7×D7
 Upper central C1 — C7

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  C73F7  C74F7

Polynomial with Galois group C7×D7 over ℚ
actionf(x)Disc(f)
14T8x14+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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