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## G = C7×Dic7order 196 = 22·72

### Direct product of C7 and Dic7

Aliases: C7×Dic7, C7⋊C28, C14.C14, C722C4, C14.4D7, C2.(C7×D7), (C7×C14).1C2, SmallGroup(196,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7×Dic7
 Chief series C1 — C7 — C14 — C7×C14 — C7×Dic7
 Lower central C7 — C7×Dic7
 Upper central C1 — C14

Generators and relations for C7×Dic7
G = < a,b,c | a7=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Permutation representations of C7×Dic7
On 28 points - transitive group 28T33
Generators in S28
(1 5 9 13 3 7 11)(2 6 10 14 4 8 12)(15 25 21 17 27 23 19)(16 26 22 18 28 24 20)
(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 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)

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

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

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

G:=TransitiveGroup(28,33);

C7×Dic7 is a maximal subgroup of   Dic72D7  C7⋊D28  C722Q8  D7×C28

70 conjugacy classes

 class 1 2 4A 4B 7A ··· 7F 7G ··· 7AA 14A ··· 14F 14G ··· 14AA 28A ··· 28L order 1 2 4 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 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 C4 C7 C14 C28 D7 Dic7 C7×D7 C7×Dic7 kernel C7×Dic7 C7×C14 C72 Dic7 C14 C7 C14 C7 C2 C1 # reps 1 1 2 6 6 12 3 3 18 18

Matrix representation of C7×Dic7 in GL2(𝔽29) generated by

 24 0 0 24
,
 22 0 0 4
,
 0 28 1 0
G:=sub<GL(2,GF(29))| [24,0,0,24],[22,0,0,4],[0,1,28,0] >;

C7×Dic7 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_7
% in TeX

G:=Group("C7xDic7");
// GroupNames label

G:=SmallGroup(196,5);
// by ID

G=gap.SmallGroup(196,5);
# by ID

G:=PCGroup([4,-2,-7,-2,-7,56,2691]);
// Polycyclic

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

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