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## G = C5×A4order 60 = 22·3·5

### Direct product of C5 and A4

Aliases: C5×A4, C22⋊C15, (C2×C10)⋊C3, SmallGroup(60,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4
 Chief series C1 — C22 — C2×C10 — C5×A4
 Lower central C22 — C5×A4
 Upper central C1 — C5

Generators and relations for C5×A4
G = < a,b,c,d | a5=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C5×A4

 class 1 2 3A 3B 5A 5B 5C 5D 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H size 1 3 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ3 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 ζ54 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ52 linear of order 5 ρ5 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 ζ5 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ53 linear of order 5 ρ6 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 ζ52 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ5 linear of order 5 ρ7 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 ζ53 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ54 linear of order 5 ρ8 1 1 ζ3 ζ32 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 ζ32ζ53 ζ3ζ5 ζ3ζ53 ζ3ζ52 ζ32ζ52 ζ32ζ54 ζ32ζ5 ζ3ζ54 linear of order 15 ρ9 1 1 ζ32 ζ3 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 ζ3ζ54 ζ32ζ53 ζ32ζ54 ζ32ζ5 ζ3ζ5 ζ3ζ52 ζ3ζ53 ζ32ζ52 linear of order 15 ρ10 1 1 ζ3 ζ32 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 ζ32ζ5 ζ3ζ52 ζ3ζ5 ζ3ζ54 ζ32ζ54 ζ32ζ53 ζ32ζ52 ζ3ζ53 linear of order 15 ρ11 1 1 ζ32 ζ3 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 ζ3ζ5 ζ32ζ52 ζ32ζ5 ζ32ζ54 ζ3ζ54 ζ3ζ53 ζ3ζ52 ζ32ζ53 linear of order 15 ρ12 1 1 ζ3 ζ32 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 ζ32ζ52 ζ3ζ54 ζ3ζ52 ζ3ζ53 ζ32ζ53 ζ32ζ5 ζ32ζ54 ζ3ζ5 linear of order 15 ρ13 1 1 ζ3 ζ32 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 ζ32ζ54 ζ3ζ53 ζ3ζ54 ζ3ζ5 ζ32ζ5 ζ32ζ52 ζ32ζ53 ζ3ζ52 linear of order 15 ρ14 1 1 ζ32 ζ3 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 ζ3ζ52 ζ32ζ54 ζ32ζ52 ζ32ζ53 ζ3ζ53 ζ3ζ5 ζ3ζ54 ζ32ζ5 linear of order 15 ρ15 1 1 ζ32 ζ3 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 ζ3ζ53 ζ32ζ5 ζ32ζ53 ζ32ζ52 ζ3ζ52 ζ3ζ54 ζ3ζ5 ζ32ζ54 linear of order 15 ρ16 3 -1 0 0 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ17 3 -1 0 0 3ζ52 3ζ53 3ζ54 3ζ5 -ζ52 -ζ54 -ζ53 -ζ5 0 0 0 0 0 0 0 0 complex faithful ρ18 3 -1 0 0 3ζ5 3ζ54 3ζ52 3ζ53 -ζ5 -ζ52 -ζ54 -ζ53 0 0 0 0 0 0 0 0 complex faithful ρ19 3 -1 0 0 3ζ53 3ζ52 3ζ5 3ζ54 -ζ53 -ζ5 -ζ52 -ζ54 0 0 0 0 0 0 0 0 complex faithful ρ20 3 -1 0 0 3ζ54 3ζ5 3ζ53 3ζ52 -ζ54 -ζ53 -ζ5 -ζ52 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C5×A4
On 20 points - transitive group 20T14
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)
(6 16 11)(7 17 12)(8 18 13)(9 19 14)(10 20 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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,16,11)(7,17,12)(8,18,13)(9,19,14)(10,20,15)>;

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

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

G:=TransitiveGroup(20,14);

On 30 points - transitive group 30T11
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 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 30)(7 26)(8 27)(9 28)(10 29)
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 26 16)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)

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

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

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

G:=TransitiveGroup(30,11);

C5×A4 is a maximal subgroup of   C5⋊S4

Matrix representation of C5×A4 in GL3(𝔽11) generated by

 3 0 0 0 3 0 0 0 3
,
 0 6 5 0 10 0 9 10 0
,
 10 0 0 2 0 1 2 1 0
,
 0 6 0 9 10 0 0 10 1
G:=sub<GL(3,GF(11))| [3,0,0,0,3,0,0,0,3],[0,0,9,6,10,10,5,0,0],[10,2,2,0,0,1,0,1,0],[0,9,0,6,10,10,0,0,1] >;

C5×A4 in GAP, Magma, Sage, TeX

C_5\times A_4
% in TeX

G:=Group("C5xA4");
// GroupNames label

G:=SmallGroup(60,9);
// by ID

G=gap.SmallGroup(60,9);
# by ID

G:=PCGroup([4,-3,-5,-2,2,362,723]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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