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## G = C5×S4order 120 = 23·3·5

### Direct product of C5 and S4

Aliases: C5×S4, A4⋊C10, C22⋊(C5×S3), (C5×A4)⋊3C2, (C2×C10)⋊1S3, SmallGroup(120,37)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×S4
G = < a,b,c,d,e | a5=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 >

Character table of C5×S4

 class 1 2A 2B 3 4 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 15A 15B 15C 15D 20A 20B 20C 20D size 1 3 6 8 6 1 1 1 1 3 3 3 3 6 6 6 6 8 8 8 8 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 -1 ζ53 ζ52 ζ54 ζ5 ζ54 ζ5 ζ52 ζ53 -ζ53 -ζ52 -ζ54 -ζ5 ζ52 ζ5 ζ54 ζ53 -ζ52 -ζ5 -ζ54 -ζ53 linear of order 10 ρ4 1 1 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ52 ζ53 ζ5 ζ54 ζ54 ζ5 ζ52 ζ53 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ54 linear of order 5 ρ5 1 1 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ54 ζ5 ζ52 ζ53 ζ53 ζ52 ζ54 ζ5 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ53 linear of order 5 ρ6 1 1 -1 1 -1 ζ54 ζ5 ζ52 ζ53 ζ52 ζ53 ζ5 ζ54 -ζ54 -ζ5 -ζ52 -ζ53 ζ5 ζ53 ζ52 ζ54 -ζ5 -ζ53 -ζ52 -ζ54 linear of order 10 ρ7 1 1 -1 1 -1 ζ52 ζ53 ζ5 ζ54 ζ5 ζ54 ζ53 ζ52 -ζ52 -ζ53 -ζ5 -ζ54 ζ53 ζ54 ζ5 ζ52 -ζ53 -ζ54 -ζ5 -ζ52 linear of order 10 ρ8 1 1 -1 1 -1 ζ5 ζ54 ζ53 ζ52 ζ53 ζ52 ζ54 ζ5 -ζ5 -ζ54 -ζ53 -ζ52 ζ54 ζ52 ζ53 ζ5 -ζ54 -ζ52 -ζ53 -ζ5 linear of order 10 ρ9 1 1 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ53 ζ52 ζ54 ζ5 ζ5 ζ54 ζ53 ζ52 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ5 linear of order 5 ρ10 1 1 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ5 ζ54 ζ53 ζ52 ζ52 ζ53 ζ5 ζ54 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ52 linear of order 5 ρ11 2 2 0 -1 0 2 2 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ12 2 2 0 -1 0 2ζ54 2ζ5 2ζ52 2ζ53 2ζ52 2ζ53 2ζ5 2ζ54 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 0 0 0 0 complex lifted from C5×S3 ρ13 2 2 0 -1 0 2ζ53 2ζ52 2ζ54 2ζ5 2ζ54 2ζ5 2ζ52 2ζ53 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 0 0 0 0 complex lifted from C5×S3 ρ14 2 2 0 -1 0 2ζ52 2ζ53 2ζ5 2ζ54 2ζ5 2ζ54 2ζ53 2ζ52 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 0 0 0 0 complex lifted from C5×S3 ρ15 2 2 0 -1 0 2ζ5 2ζ54 2ζ53 2ζ52 2ζ53 2ζ52 2ζ54 2ζ5 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 0 0 0 0 complex lifted from C5×S3 ρ16 3 -1 -1 0 1 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from S4 ρ17 3 -1 1 0 -1 3 3 3 3 -1 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ18 3 -1 -1 0 1 3ζ54 3ζ5 3ζ52 3ζ53 -ζ52 -ζ53 -ζ5 -ζ54 -ζ54 -ζ5 -ζ52 -ζ53 0 0 0 0 ζ5 ζ53 ζ52 ζ54 complex faithful ρ19 3 -1 -1 0 1 3ζ53 3ζ52 3ζ54 3ζ5 -ζ54 -ζ5 -ζ52 -ζ53 -ζ53 -ζ52 -ζ54 -ζ5 0 0 0 0 ζ52 ζ5 ζ54 ζ53 complex faithful ρ20 3 -1 -1 0 1 3ζ5 3ζ54 3ζ53 3ζ52 -ζ53 -ζ52 -ζ54 -ζ5 -ζ5 -ζ54 -ζ53 -ζ52 0 0 0 0 ζ54 ζ52 ζ53 ζ5 complex faithful ρ21 3 -1 1 0 -1 3ζ5 3ζ54 3ζ53 3ζ52 -ζ53 -ζ52 -ζ54 -ζ5 ζ5 ζ54 ζ53 ζ52 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 complex faithful ρ22 3 -1 1 0 -1 3ζ53 3ζ52 3ζ54 3ζ5 -ζ54 -ζ5 -ζ52 -ζ53 ζ53 ζ52 ζ54 ζ5 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 complex faithful ρ23 3 -1 -1 0 1 3ζ52 3ζ53 3ζ5 3ζ54 -ζ5 -ζ54 -ζ53 -ζ52 -ζ52 -ζ53 -ζ5 -ζ54 0 0 0 0 ζ53 ζ54 ζ5 ζ52 complex faithful ρ24 3 -1 1 0 -1 3ζ54 3ζ5 3ζ52 3ζ53 -ζ52 -ζ53 -ζ5 -ζ54 ζ54 ζ5 ζ52 ζ53 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 complex faithful ρ25 3 -1 1 0 -1 3ζ52 3ζ53 3ζ5 3ζ54 -ζ5 -ζ54 -ζ53 -ζ52 ζ52 ζ53 ζ5 ζ54 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 complex faithful

Permutation representations of C5×S4
On 20 points - transitive group 20T34
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)
(6 11)(7 12)(8 13)(9 14)(10 15)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (6,11)(7,12)(8,13)(9,14)(10,15)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (6,11)(7,12)(8,13)(9,14)(10,15) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15)], [(6,11,16),(7,12,17),(8,13,18),(9,14,19),(10,15,20)], [(6,11),(7,12),(8,13),(9,14),(10,15)])

G:=TransitiveGroup(20,34);

On 30 points - transitive group 30T33
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 17)(2 18)(3 19)(4 20)(5 16)(11 27)(12 28)(13 29)(14 30)(15 26)
(6 24)(7 25)(8 21)(9 22)(10 23)(11 27)(12 28)(13 29)(14 30)(15 26)
(1 15 6)(2 11 7)(3 12 8)(4 13 9)(5 14 10)(16 30 23)(17 26 24)(18 27 25)(19 28 21)(20 29 22)
(6 15)(7 11)(8 12)(9 13)(10 14)(21 28)(22 29)(23 30)(24 26)(25 27)

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

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

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

G:=TransitiveGroup(30,33);

On 30 points - transitive group 30T34
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 11)(2 12)(3 13)(4 14)(5 15)(6 28)(7 29)(8 30)(9 26)(10 27)
(6 28)(7 29)(8 30)(9 26)(10 27)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 9 18)(2 10 19)(3 6 20)(4 7 16)(5 8 17)(11 26 24)(12 27 25)(13 28 21)(14 29 22)(15 30 23)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 21)(7 22)(8 23)(9 24)(10 25)(16 29)(17 30)(18 26)(19 27)(20 28)

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

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

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

G:=TransitiveGroup(30,34);

Matrix representation of C5×S4 in GL3(𝔽11) generated by

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

C5×S4 in GAP, Magma, Sage, TeX

C_5\times S_4
% in TeX

G:=Group("C5xS4");
// GroupNames label

G:=SmallGroup(120,37);
// by ID

G=gap.SmallGroup(120,37);
# by ID

G:=PCGroup([5,-2,-5,-3,-2,2,302,1203,133,754,239]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=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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