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G = C10xS4  order 240 = 24·3·5

Direct product of C10 and S4

direct product, non-abelian, soluble, monomial

Aliases: C10xS4, (C2xA4):C10, A4:(C2xC10), C23:(C5xS3), (C2xC10):2D6, C22:(S3xC10), (C10xA4):3C2, (C5xA4):4C22, (C22xC10):1S3, SmallGroup(240,196)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — A4 — C10xS4
Chief series
C1 — C22 — A4 — C5xA4 — C5xS4 — C10xS4
Lower central
A4 — C10xS4
Upper central
C1 — C10

Generators and relations for C10xS4
 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, C2xC4, D4, C23, C23, C10, C10, A4, D6, C15, C2xD4, C20, C2xC10, C2xC10, S4, C2xA4, C5xS3, C30, C2xC20, C5xD4, C22xC10, C22xC10, C2xS4, C5xA4, S3xC10, D4xC10, C5xS4, C10xA4, C10xS4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2xC10, S4, C5xS3, C2xS4, S3xC10, C5xS4, C10xS4

Permutation representations of C10xS4
►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);
 

C10xS4 is a maximal subgroup of   Dic5:S4  D10:S4

50 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B5A5B5C5D 6 10A10B10C10D10E···10L10M···10T15A15B15C15D20A···20H30A30B30C30D
order122222344555561010101010···1010···101515151520···2030303030
size1133668661111811113···36···688886···68888

50 irreducible representations

dim11111122223333
type+++++++
imageC1C2C2C5C10C10S3D6C5xS3S3xC10S4C2xS4C5xS4C10xS4
kernelC10xS4C5xS4C10xA4C2xS4S4C2xA4C22xC10C2xC10C23C22C10C5C2C1
# reps12148411442288

Matrix representation of C10xS4 ►in GL3(F11) generated by

200
020
002
,
1000
010
0010
,
1000
0100
001
,
008
200
090
,
050
900
0010
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] >;
 

C10xS4 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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