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G = D7×C14order 196 = 22·72

Direct product of C14 and D7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C14
 Chief series C1 — C7 — C72 — C7×D7 — D7×C14
 Lower central C7 — D7×C14
 Upper central C1 — C14

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

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

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

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

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

G:=TransitiveGroup(28,34);

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

70 conjugacy classes

 class 1 2A 2B 2C 7A ··· 7F 7G ··· 7AA 14A ··· 14F 14G ··· 14AA 14AB ··· 14AM order 1 2 2 2 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 size 1 1 7 7 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 7 ··· 7

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 D7 D14 C7×D7 D7×C14 kernel D7×C14 C7×D7 C7×C14 D14 D7 C14 C14 C7 C2 C1 # reps 1 2 1 6 12 6 3 3 18 18

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

 5 0 0 5
,
 16 0 0 20
,
 0 20 16 0
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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