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Cayley diagram

C5×SL(2,ℤ/4) #6743

order240
isoclinismisoclinic class of S4
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = S4 · Out(G) = C2×C4
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×SL(2,ℤ/4)
unambiguous semidirect productA4⋊(C4×C5)

Splittings

derivedA4 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5S4
commutator, hypocenterA4C4×C5
FrattiniC2C5×S4
Fitting, socleC23×C5D3
prime radicalC2×C5×A4C2
cosocleC2×A4C2×C5
LCSG ▷ A4
UCSC1 ◁ C2×C5

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2,3,4)(5,6,8,7)(9,10,11,12,13);(6,7,8)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#91899, #91901, D5×SL(2,ℤ/4), …
maximal normal subgroup of#91899, #91901, D5×SL(2,ℤ/4), …
maximal quotient ofC5×A4⋊C8, C5×#859, C5×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 104);