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## G = C14×S4order 336 = 24·3·7

### Direct product of C14 and S4

Aliases: C14×S4, (C2×A4)⋊C14, A4⋊(C2×C14), (C2×C14)⋊2D6, C22⋊(S3×C14), (A4×C14)⋊3C2, C232(S3×C7), (C7×A4)⋊4C22, (C22×C14)⋊1S3, SmallGroup(336,214)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C14×S4
 Chief series C1 — C22 — A4 — C7×A4 — C7×S4 — C14×S4
 Lower central A4 — C14×S4
 Upper central C1 — C14

Generators and relations for C14×S4
G = < a,b,c,d,e | a14=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 >

Subgroups: 196 in 66 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, D4, C23, C23, A4, D6, C14, C14, C2×D4, C21, S4, C2×A4, C28, C2×C14, C2×C14, S3×C7, C42, C2×S4, C2×C28, C7×D4, C22×C14, C22×C14, C7×A4, S3×C14, D4×C14, C7×S4, A4×C14, C14×S4
Quotients: C1, C2, C22, S3, C7, D6, C14, S4, C2×C14, S3×C7, C2×S4, S3×C14, C7×S4, C14×S4

Smallest permutation representation of C14×S4
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)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 20 39)(2 21 40)(3 22 41)(4 23 42)(5 24 29)(6 25 30)(7 26 31)(8 27 32)(9 28 33)(10 15 34)(11 16 35)(12 17 36)(13 18 37)(14 19 38)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 41)(16 42)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 6 7A ··· 7F 14A ··· 14F 14G ··· 14R 14S ··· 14AD 21A ··· 21F 28A ··· 28L 42A ··· 42F order 1 2 2 2 2 2 3 4 4 6 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 3 3 6 6 8 6 6 8 1 ··· 1 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 S4 C2×S4 C7×S4 C14×S4 kernel C14×S4 C7×S4 A4×C14 C2×S4 S4 C2×A4 C22×C14 C2×C14 C23 C22 C14 C7 C2 C1 # reps 1 2 1 6 12 6 1 1 6 6 2 2 12 12

Matrix representation of C14×S4 in GL3(𝔽337) generated by

 162 0 0 0 162 0 0 0 162
,
 336 0 0 0 336 0 0 0 1
,
 1 0 0 0 336 0 0 0 336
,
 0 0 1 1 0 0 0 1 0
,
 336 0 0 0 0 336 0 336 0
G:=sub<GL(3,GF(337))| [162,0,0,0,162,0,0,0,162],[336,0,0,0,336,0,0,0,1],[1,0,0,0,336,0,0,0,336],[0,1,0,0,0,1,1,0,0],[336,0,0,0,0,336,0,336,0] >;

C14×S4 in GAP, Magma, Sage, TeX

C_{14}\times S_4
% in TeX

G:=Group("C14xS4");
// GroupNames label

G:=SmallGroup(336,214);
// by ID

G=gap.SmallGroup(336,214);
# by ID

G:=PCGroup([6,-2,-2,-7,-3,-2,2,1347,5044,202,3029,347]);
// Polycyclic

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