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G = C7×D4order 56 = 23·7

Direct product of C7 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×D4, C4⋊C14, C283C2, C22⋊C14, C14.6C22, (C2×C14)⋊1C2, C2.1(C2×C14), SmallGroup(56,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×D4
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4
 Lower central C1 — C2 — C7×D4
 Upper central C1 — C14 — C7×D4

Generators and relations for C7×D4
G = < a,b,c | a7=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Permutation representations of C7×D4
On 28 points - transitive group 28T5
Generators in S28
(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)
(1 12 19 24)(2 13 20 25)(3 14 21 26)(4 8 15 27)(5 9 16 28)(6 10 17 22)(7 11 18 23)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)

G:=sub<Sym(28)| (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), (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)>;

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), (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21) );

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)], [(1,12,19,24),(2,13,20,25),(3,14,21,26),(4,8,15,27),(5,9,16,28),(6,10,17,22),(7,11,18,23)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)])

G:=TransitiveGroup(28,5);

35 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7F 14A ··· 14F 14G ··· 14R 28A ··· 28F order 1 2 2 2 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C7 C14 C14 D4 C7×D4 kernel C7×D4 C28 C2×C14 D4 C4 C22 C7 C1 # reps 1 1 2 6 6 12 1 6

Matrix representation of C7×D4 in GL2(𝔽29) generated by

 24 0 0 24
,
 28 1 27 1
,
 1 28 0 28
G:=sub<GL(2,GF(29))| [24,0,0,24],[28,27,1,1],[1,0,28,28] >;

C7×D4 in GAP, Magma, Sage, TeX

C_7\times D_4
% in TeX

G:=Group("C7xD4");
// GroupNames label

G:=SmallGroup(56,9);
// by ID

G=gap.SmallGroup(56,9);
# by ID

G:=PCGroup([4,-2,-2,-7,-2,241]);
// Polycyclic

G:=Group<a,b,c|a^7=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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