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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

C1C2 — C7×D4
C1C2C14C2×C14 — C7×D4
C1C2 — C7×D4
C1C14 — C7×D4

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

2C2
2C2
2C14
2C14

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

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

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

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

G:=TransitiveGroup(28,5);

35 conjugacy classes

class 1 2A2B2C 4 7A···7F14A···14F14G···14R28A···28F
order122247···714···1414···1428···28
size112221···11···12···22···2

35 irreducible representations

dim11111122
type++++
imageC1C2C2C7C14C14D4C7×D4
kernelC7×D4C28C2×C14D4C4C22C7C1
# reps112661216

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

240
024
,
281
271
,
128
028
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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