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## G = C10×S4order 240 = 24·3·5

### Direct product of C10 and S4

Aliases: C10×S4, (C2×A4)⋊C10, A4⋊(C2×C10), C23⋊(C5×S3), (C2×C10)⋊2D6, C22⋊(S3×C10), (C10×A4)⋊3C2, (C5×A4)⋊4C22, (C22×C10)⋊1S3, SmallGroup(240,196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C10×S4
 Chief series C1 — C22 — A4 — C5×A4 — C5×S4 — C10×S4
 Lower central A4 — C10×S4
 Upper central C1 — C10

Generators and relations for C10×S4
G = < a,b,c,d,e | a10=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, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, A4, D6, C15, C2×D4, C20, C2×C10, C2×C10, S4, C2×A4, C5×S3, C30, C2×C20, C5×D4, C22×C10, C22×C10, C2×S4, C5×A4, S3×C10, D4×C10, C5×S4, C10×A4, C10×S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C5×S4, C10×S4

Permutation representations of C10×S4
On 30 points - transitive group 30T65
Generators in S30
(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)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 28 17)(2 29 18)(3 30 19)(4 21 20)(5 22 11)(6 23 12)(7 24 13)(8 25 14)(9 26 15)(10 27 16)
(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)

G:=sub<Sym(30)| (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), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,28,17)(2,29,18)(3,30,19)(4,21,20)(5,22,11)(6,23,12)(7,24,13)(8,25,14)(9,26,15)(10,27,16), (11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)>;

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), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,28,17)(2,29,18)(3,30,19)(4,21,20)(5,22,11)(6,23,12)(7,24,13)(8,25,14)(9,26,15)(10,27,16), (11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21) );

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)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,28,17),(2,29,18),(3,30,19),(4,21,20),(5,22,11),(6,23,12),(7,24,13),(8,25,14),(9,26,15),(10,27,16)], [(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,21)]])

G:=TransitiveGroup(30,65);

C10×S4 is a maximal subgroup of   Dic5⋊S4  D10⋊S4

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10L 10M ··· 10T 15A 15B 15C 15D 20A ··· 20H 30A 30B 30C 30D order 1 2 2 2 2 2 3 4 4 5 5 5 5 6 10 10 10 10 10 ··· 10 10 ··· 10 15 15 15 15 20 ··· 20 30 30 30 30 size 1 1 3 3 6 6 8 6 6 1 1 1 1 8 1 1 1 1 3 ··· 3 6 ··· 6 8 8 8 8 6 ··· 6 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 S4 C2×S4 C5×S4 C10×S4 kernel C10×S4 C5×S4 C10×A4 C2×S4 S4 C2×A4 C22×C10 C2×C10 C23 C22 C10 C5 C2 C1 # reps 1 2 1 4 8 4 1 1 4 4 2 2 8 8

Matrix representation of C10×S4 in GL3(𝔽11) generated by

 2 0 0 0 2 0 0 0 2
,
 10 0 0 0 1 0 0 0 10
,
 10 0 0 0 10 0 0 0 1
,
 0 0 8 2 0 0 0 9 0
,
 0 5 0 9 0 0 0 0 10
G:=sub<GL(3,GF(11))| [2,0,0,0,2,0,0,0,2],[10,0,0,0,1,0,0,0,10],[10,0,0,0,10,0,0,0,1],[0,2,0,0,0,9,8,0,0],[0,9,0,5,0,0,0,0,10] >;

C10×S4 in GAP, Magma, Sage, TeX

C_{10}\times S_4
% in TeX

G:=Group("C10xS4");
// GroupNames label

G:=SmallGroup(240,196);
// by ID

G=gap.SmallGroup(240,196);
# by ID

G:=PCGroup([6,-2,-2,-5,-3,-2,2,963,3604,202,2165,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=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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