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## G = A4×C14order 168 = 23·3·7

### Direct product of C14 and A4

Aliases: A4×C14, C22⋊C42, C232C21, (C2×C14)⋊6C6, (C22×C14)⋊1C3, SmallGroup(168,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C14
 Chief series C1 — C22 — C2×C14 — C7×A4 — A4×C14
 Lower central C22 — A4×C14
 Upper central C1 — C14

Generators and relations for A4×C14
G = < a,b,c,d | a14=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C14
On 42 points
Generators in S42
(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)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 35 27)(2 36 28)(3 37 15)(4 38 16)(5 39 17)(6 40 18)(7 41 19)(8 42 20)(9 29 21)(10 30 22)(11 31 23)(12 32 24)(13 33 25)(14 34 26)

G:=sub<Sym(42)| (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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,35,27)(2,36,28)(3,37,15)(4,38,16)(5,39,17)(6,40,18)(7,41,19)(8,42,20)(9,29,21)(10,30,22)(11,31,23)(12,32,24)(13,33,25)(14,34,26)>;

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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,35,27)(2,36,28)(3,37,15)(4,38,16)(5,39,17)(6,40,18)(7,41,19)(8,42,20)(9,29,21)(10,30,22)(11,31,23)(12,32,24)(13,33,25)(14,34,26) );

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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,35,27),(2,36,28),(3,37,15),(4,38,16),(5,39,17),(6,40,18),(7,41,19),(8,42,20),(9,29,21),(10,30,22),(11,31,23),(12,32,24),(13,33,25),(14,34,26)]])

A4×C14 is a maximal subgroup of   A4⋊Dic7

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 7A ··· 7F 14A ··· 14F 14G ··· 14R 21A ··· 21L 42A ··· 42L order 1 2 2 2 3 3 6 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 3 3 4 4 4 4 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C7 C14 C21 C42 A4 C2×A4 C7×A4 A4×C14 kernel A4×C14 C7×A4 C22×C14 C2×C14 C2×A4 A4 C23 C22 C14 C7 C2 C1 # reps 1 1 2 2 6 6 12 12 1 1 6 6

Matrix representation of A4×C14 in GL3(𝔽43) generated by

 22 0 0 0 22 0 0 0 22
,
 42 0 0 0 42 0 1 0 1
,
 42 0 0 11 1 0 0 0 42
,
 32 41 0 18 11 1 5 1 0
G:=sub<GL(3,GF(43))| [22,0,0,0,22,0,0,0,22],[42,0,1,0,42,0,0,0,1],[42,11,0,0,1,0,0,0,42],[32,18,5,41,11,1,0,1,0] >;

A4×C14 in GAP, Magma, Sage, TeX

A_4\times C_{14}
% in TeX

G:=Group("A4xC14");
// GroupNames label

G:=SmallGroup(168,52);
// by ID

G=gap.SmallGroup(168,52);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,2,848,1584]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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