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G = C5×A4  order 60 = 22·3·5

Direct product of C5 and A4

direct product, metabelian, soluble, monomial, A-group

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 >

C1
3C2
4C3
C5
C22
3C10
4C15
A4
C2×C10
C5×A4

Character table of C5×A4

 class 123A3B5A5B5C5D10A10B10C10D15A15B15C15D15E15F15G15H
 size 13441111333344444444
ρ111111111111111111111    trivial
ρ211ζ32ζ311111111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ311ζ3ζ3211111111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ41111ζ52ζ53ζ54ζ5ζ52ζ54ζ53ζ5ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ52    linear of order 5
ρ51111ζ53ζ52ζ5ζ54ζ53ζ5ζ52ζ54ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ53    linear of order 5
ρ61111ζ5ζ54ζ52ζ53ζ5ζ52ζ54ζ53ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ5    linear of order 5
ρ71111ζ54ζ5ζ53ζ52ζ54ζ53ζ5ζ52ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ54    linear of order 5
ρ811ζ3ζ32ζ54ζ5ζ53ζ52ζ54ζ53ζ5ζ52ζ32ζ53ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ3ζ54    linear of order 15
ρ911ζ32ζ3ζ52ζ53ζ54ζ5ζ52ζ54ζ53ζ5ζ3ζ54ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ32ζ52    linear of order 15
ρ1011ζ3ζ32ζ53ζ52ζ5ζ54ζ53ζ5ζ52ζ54ζ32ζ5ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ3ζ53    linear of order 15
ρ1111ζ32ζ3ζ53ζ52ζ5ζ54ζ53ζ5ζ52ζ54ζ3ζ5ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ32ζ53    linear of order 15
ρ1211ζ3ζ32ζ5ζ54ζ52ζ53ζ5ζ52ζ54ζ53ζ32ζ52ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ3ζ5    linear of order 15
ρ1311ζ3ζ32ζ52ζ53ζ54ζ5ζ52ζ54ζ53ζ5ζ32ζ54ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ3ζ52    linear of order 15
ρ1411ζ32ζ3ζ5ζ54ζ52ζ53ζ5ζ52ζ54ζ53ζ3ζ52ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ32ζ5    linear of order 15
ρ1511ζ32ζ3ζ54ζ5ζ53ζ52ζ54ζ53ζ5ζ52ζ3ζ53ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ32ζ54    linear of order 15
ρ163-1003333-1-1-1-100000000    orthogonal lifted from A4
ρ173-1003ζ523ζ533ζ543ζ5-ζ52-ζ54-ζ53-ζ500000000    complex faithful
ρ183-1003ζ53ζ543ζ523ζ53-ζ5-ζ52-ζ54-ζ5300000000    complex faithful
ρ193-1003ζ533ζ523ζ53ζ54-ζ53-ζ5-ζ52-ζ5400000000    complex faithful
ρ203-1003ζ543ζ53ζ533ζ52-ζ54-ζ53-ζ5-ζ5200000000    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

300
030
003
,
065
0100
9100
,
1000
201
210
,
060
9100
0101
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
 

Export

Subgroup lattice of C5×A4 in TeX
Character table of C5×A4 in TeX

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