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G = D7×C14order 196 = 22·72

Direct product of C14 and D7

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: D7×C14, C14⋊C14, C722C22, C7⋊(C2×C14), (C7×C14)⋊1C2, SmallGroup(196,10)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C14
C1C7C72C7×D7 — D7×C14
C7 — D7×C14
C1C14

Generators and relations for D7×C14
 G = < a,b,c | a14=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
2C7
2C7
2C7
7C22
2C14
2C14
2C14
7C14
7C14
7C2×C14

Permutation representations of D7×C14
On 28 points - transitive group 28T34
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 13 11 9 7 5 3)(2 14 12 10 8 6 4)(15 17 19 21 23 25 27)(16 18 20 22 24 26 28)
(1 25)(2 26)(3 27)(4 28)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)

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

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

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

G:=TransitiveGroup(28,34);

D7×C14 is a maximal subgroup of   C722D4  C7⋊D28

70 conjugacy classes

class 1 2A2B2C7A···7F7G···7AA14A···14F14G···14AA14AB···14AM
order12227···77···714···1414···1414···14
size11771···12···21···12···27···7

70 irreducible representations

dim1111112222
type+++++
imageC1C2C2C7C14C14D7D14C7×D7D7×C14
kernelD7×C14C7×D7C7×C14D14D7C14C14C7C2C1
# reps1216126331818

Matrix representation of D7×C14 in GL2(𝔽29) generated by

50
05
,
160
020
,
020
160
G:=sub<GL(2,GF(29))| [5,0,0,5],[16,0,0,20],[0,16,20,0] >;

D7×C14 in GAP, Magma, Sage, TeX

D_7\times C_{14}
% in TeX

G:=Group("D7xC14");
// GroupNames label

G:=SmallGroup(196,10);
// by ID

G=gap.SmallGroup(196,10);
# by ID

G:=PCGroup([4,-2,-2,-7,-7,2691]);
// Polycyclic

G:=Group<a,b,c|a^14=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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Subgroup lattice of D7×C14 in TeX

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