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## G = C14×Dic7order 392 = 23·72

### Direct product of C14 and Dic7

Aliases: C14×Dic7, C14⋊C28, C14.20D14, C142.1C2, C72(C2×C28), (C7×C14)⋊2C4, C726(C2×C4), C22.(C7×D7), (C2×C14).5D7, C2.2(D7×C14), (C2×C14).2C14, C14.4(C2×C14), (C7×C14).9C22, SmallGroup(392,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C14×Dic7
 Chief series C1 — C7 — C14 — C7×C14 — C7×Dic7 — C14×Dic7
 Lower central C7 — C14×Dic7
 Upper central C1 — C2×C14

Generators and relations for C14×Dic7
G = < a,b,c | a14=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C14×Dic7
On 56 points
Generators in S56
(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)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 44 9 52 3 46 11 54 5 48 13 56 7 50)(2 45 10 53 4 47 12 55 6 49 14 43 8 51)(15 32 21 38 27 30 19 36 25 42 17 34 23 40)(16 33 22 39 28 31 20 37 26 29 18 35 24 41)
(1 16 54 37)(2 17 55 38)(3 18 56 39)(4 19 43 40)(5 20 44 41)(6 21 45 42)(7 22 46 29)(8 23 47 30)(9 24 48 31)(10 25 49 32)(11 26 50 33)(12 27 51 34)(13 28 52 35)(14 15 53 36)

G:=sub<Sym(56)| (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36)>;

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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36) );

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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,44,9,52,3,46,11,54,5,48,13,56,7,50),(2,45,10,53,4,47,12,55,6,49,14,43,8,51),(15,32,21,38,27,30,19,36,25,42,17,34,23,40),(16,33,22,39,28,31,20,37,26,29,18,35,24,41)], [(1,16,54,37),(2,17,55,38),(3,18,56,39),(4,19,43,40),(5,20,44,41),(6,21,45,42),(7,22,46,29),(8,23,47,30),(9,24,48,31),(10,25,49,32),(11,26,50,33),(12,27,51,34),(13,28,52,35),(14,15,53,36)]])

140 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 7G ··· 7AA 14A ··· 14R 14S ··· 14CC 28A ··· 28X order 1 2 2 2 4 4 4 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 7 7 7 7 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 7 ··· 7

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C7 C14 C14 C28 D7 Dic7 D14 C7×D7 C7×Dic7 D7×C14 kernel C14×Dic7 C7×Dic7 C142 C7×C14 C2×Dic7 Dic7 C2×C14 C14 C2×C14 C14 C14 C22 C2 C2 # reps 1 2 1 4 6 12 6 24 3 6 3 18 36 18

Matrix representation of C14×Dic7 in GL3(𝔽29) generated by

 20 0 0 0 22 0 0 0 22
,
 28 0 0 0 23 0 0 0 24
,
 17 0 0 0 0 28 0 28 0
G:=sub<GL(3,GF(29))| [20,0,0,0,22,0,0,0,22],[28,0,0,0,23,0,0,0,24],[17,0,0,0,0,28,0,28,0] >;

C14×Dic7 in GAP, Magma, Sage, TeX

C_{14}\times {\rm Dic}_7
% in TeX

G:=Group("C14xDic7");
// GroupNames label

G:=SmallGroup(392,26);
// by ID

G=gap.SmallGroup(392,26);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,140,8404]);
// Polycyclic

G:=Group<a,b,c|a^14=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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