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## G = C7×S4order 168 = 23·3·7

### Direct product of C7 and S4

Aliases: C7×S4, A4⋊C14, C22⋊(S3×C7), (C7×A4)⋊3C2, (C2×C14)⋊1S3, SmallGroup(168,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C7×S4
 Chief series C1 — C22 — A4 — C7×A4 — C7×S4
 Lower central A4 — C7×S4
 Upper central C1 — C7

Generators and relations for C7×S4
G = < a,b,c,d,e | a7=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Permutation representations of C7×S4
On 28 points - transitive group 28T31
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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 26)(16 27)(17 28)(18 22)(19 23)(20 24)(21 25)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)
(8 24 20)(9 25 21)(10 26 15)(11 27 16)(12 28 17)(13 22 18)(14 23 19)
(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)

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

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

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

G:=TransitiveGroup(28,31);

35 conjugacy classes

 class 1 2A 2B 3 4 7A ··· 7F 14A ··· 14F 14G ··· 14L 21A ··· 21F 28A ··· 28F order 1 2 2 3 4 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 size 1 3 6 8 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

35 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + + image C1 C2 C7 C14 S3 S3×C7 S4 C7×S4 kernel C7×S4 C7×A4 S4 A4 C2×C14 C22 C7 C1 # reps 1 1 6 6 1 6 2 12

Matrix representation of C7×S4 in GL3(𝔽337) generated by

 79 0 0 0 79 0 0 0 79
,
 0 0 1 336 336 336 1 0 0
,
 336 336 336 0 0 1 0 1 0
,
 1 0 0 336 336 336 0 1 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(337))| [79,0,0,0,79,0,0,0,79],[0,336,1,0,336,0,1,336,0],[336,0,0,336,0,1,336,1,0],[1,336,0,0,336,1,0,336,0],[1,0,0,0,0,1,0,1,0] >;

C7×S4 in GAP, Magma, Sage, TeX

C_7\times S_4
% in TeX

G:=Group("C7xS4");
// GroupNames label

G:=SmallGroup(168,45);
// by ID

G=gap.SmallGroup(168,45);
# by ID

G:=PCGroup([5,-2,-7,-3,-2,2,422,1683,133,1054,239]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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