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G = C4×A5order 240 = 24·3·5

Direct product of C4 and A5

direct product, non-abelian, not soluble, A-group

Aliases: C4×A5, C2.1(C2×A5), (C2×A5).2C2, SmallGroup(240,92)

Series: ChiefDerived Lower central Upper central

C1C2C4 — C4×A5
A5 — C4×A5
A5 — C4×A5
C1C4

15C2
15C2
10C3
6C5
5C22
15C4
15C22
15C22
10S3
10C6
10S3
6C10
6D5
6D5
5C23
15C2×C4
15C2×C4
5A4
10C12
10D6
10Dic3
6Dic5
6C20
6D10
5C22×C4
5C2×A4
10C4×S3
6C4×D5
5C4×A4

Character table of C4×A5

 class 12A2B2C34A4B4C4D5A5B610A10B12A12B20A20B20C20D
 size 111515201115151212201212202012121212
ρ111111111111111111111    trivial
ρ211111-1-1-1-111111-1-1-1-1-1-1    linear of order 2
ρ31-1-111-ii-ii11-1-1-1i-i-iii-i    linear of order 4
ρ41-1-111i-ii-i11-1-1-1-iii-i-ii    linear of order 4
ρ533-1-1033-1-11+5/21-5/201-5/21+5/2001+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ633-1-1033-1-11-5/21+5/201+5/21-5/2001-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ733-1-10-3-3111-5/21+5/201+5/21-5/200-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from C2×A5
ρ833-1-10-3-3111+5/21-5/201-5/21+5/200-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from C2×A5
ρ93-31-103i-3i-ii1-5/21+5/20-1-5/2-1+5/200ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ52    complex faithful
ρ103-31-10-3i3ii-i1+5/21-5/20-1+5/2-1-5/200ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ5    complex faithful
ρ113-31-103i-3i-ii1+5/21-5/20-1+5/2-1-5/200ζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ5    complex faithful
ρ123-31-10-3i3ii-i1-5/21+5/20-1-5/2-1+5/200ζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ52    complex faithful
ρ1344001-4-400-1-11-1-1-1-11111    orthogonal lifted from C2×A5
ρ14440014400-1-11-1-111-1-1-1-1    orthogonal lifted from A5
ρ154-40014i-4i00-1-1-111-ii-iii-i    complex faithful
ρ164-4001-4i4i00-1-1-111i-ii-i-ii    complex faithful
ρ175511-1-5-5-1-100-100110000    orthogonal lifted from C2×A5
ρ185511-1551100-100-1-10000    orthogonal lifted from A5
ρ195-5-11-1-5i5i-ii00100-ii0000    complex faithful
ρ205-5-11-15i-5ii-i00100i-i0000    complex faithful

Permutation representations of C4×A5
On 20 points - transitive group 20T63
Generators in S20
(1 18 11 8)(2 19 12 9)(3 16 13 6)(4 17 14 7)(5 10 15 20)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,63);

On 24 points - transitive group 24T574
Generators in S24
(1 24 3 14)(2 9 4 19)(5 20 15 10)(6 7 16 17)(8 23 18 13)(11 12 21 22)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)

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

G:=Group( (1,24,3,14)(2,9,4,19)(5,20,15,10)(6,7,16,17)(8,23,18,13)(11,12,21,22), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,24,3,14),(2,9,4,19),(5,20,15,10),(6,7,16,17),(8,23,18,13),(11,12,21,22)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,574);

On 24 points - transitive group 24T575
Generators in S24
(1 6 3 16)(2 11 4 21)(5 10 15 20)(7 12 17 22)(8 9 18 19)(13 14 23 24)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)

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

G:=Group( (1,6,3,16)(2,11,4,21)(5,10,15,20)(7,12,17,22)(8,9,18,19)(13,14,23,24), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,6,3,16),(2,11,4,21),(5,10,15,20),(7,12,17,22),(8,9,18,19),(13,14,23,24)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,575);

C4×A5 is a maximal subgroup of   A5⋊C8  C4⋊S5  A5⋊Q8
C4×A5 is a maximal quotient of   C8.A5

Matrix representation of C4×A5 in GL3(𝔽5) generated by

444
302
414
,
332
043
234
G:=sub<GL(3,GF(5))| [4,3,4,4,0,1,4,2,4],[3,0,2,3,4,3,2,3,4] >;

C4×A5 in GAP, Magma, Sage, TeX

C_4\times A_5
% in TeX

G:=Group("C4xA5");
// GroupNames label

G:=SmallGroup(240,92);
// by ID

G=gap.SmallGroup(240,92);
# by ID

Export

Subgroup lattice of C4×A5 in TeX
Character table of C4×A5 in TeX

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