Copied to
clipboard

G = C4.A5order 240 = 24·3·5

The central extension by C4 of A5

non-abelian, not soluble

Aliases: C4.A5, SL2(𝔽5)⋊2C2, C2.2(C2×A5), SmallGroup(240,93)

Series: ChiefDerived Lower central Upper central

C1C2C4 — C4.A5
SL2(𝔽5) — C4.A5
SL2(𝔽5) — C4.A5
C1C4

30C2
10C3
6C5
15C22
15C4
10C6
10S3
10S3
6C10
6D5
6D5
5Q8
15D4
15C2×C4
10D6
10Dic3
10C12
6C20
6Dic5
6D10
5C4○D4
5SL2(𝔽3)
10C4×S3
6C4×D5
5C4.A4

Character table of C4.A5

 class 12A2B34A4B4C5A5B610A10B12A12B20A20B20C20D
 size 11302011301212201212202012121212
ρ1111111111111111111    trivial
ρ211-11-1-1111111-1-1-1-1-1-1    linear of order 2
ρ32-20-1-2i2i0-1-5/2-1+5/211+5/21-5/2i-iζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ52    complex faithful
ρ42-20-1-2i2i0-1+5/2-1-5/211-5/21+5/2i-iζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ5    complex faithful
ρ52-20-12i-2i0-1+5/2-1-5/211-5/21+5/2-iiζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ5    complex faithful
ρ62-20-12i-2i0-1-5/2-1+5/211+5/21-5/2-iiζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ52    complex faithful
ρ733-1033-11-5/21+5/201-5/21+5/2001+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ833-1033-11+5/21-5/201+5/21-5/2001-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ93310-3-3-11+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
ρ103310-3-3-11-5/21+5/201-5/21+5/200-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from C2×A5
ρ114401-4-40-1-11-1-1-1-11111    orthogonal lifted from C2×A5
ρ124401440-1-11-1-111-1-1-1-1    orthogonal lifted from A5
ρ134-4014i-4i0-1-1-111i-i-iii-i    complex faithful
ρ144-401-4i4i0-1-1-111-iii-i-ii    complex faithful
ρ1555-1-1-5-5100-100110000    orthogonal lifted from C2×A5
ρ16551-155100-100-1-10000    orthogonal lifted from A5
ρ176-6006i-6i0110-1-100i-i-ii    complex faithful
ρ186-600-6i6i0110-1-100-iii-i    complex faithful

Permutation representations of C4.A5
On 24 points - transitive group 24T576
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 8)(2 18)(3 11)(4 22)(5 20)(6 15)(7 10)(13 16)(14 24)(17 21)

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

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

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

G:=TransitiveGroup(24,576);

C4.A5 is a maximal subgroup of   GL2(𝔽5)  C8.A5  C4.6S5  C4.S5  C4.3S5  D4.A5  Q8.A5
C4.A5 is a maximal quotient of   C4×SL2(𝔽5)

Matrix representation of C4.A5 in GL2(𝔽5) generated by

43
23
,
31
22
G:=sub<GL(2,GF(5))| [4,2,3,3],[3,2,1,2] >;

C4.A5 in GAP, Magma, Sage, TeX

C_4.A_5
% in TeX

G:=Group("C4.A5");
// GroupNames label

G:=SmallGroup(240,93);
// by ID

G=gap.SmallGroup(240,93);
# by ID

Export

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

׿
×
𝔽