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Cayley diagram

D5×S4 #6833

order240
isoclinismisoclinic class of D5×S4
automorphismAut(G) = AGL(1,𝔽5)×S4 (order 480) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

direct productD5×S4
unambiguous semidirect product(C5×S4)⋊C2
unambiguous semidirect product(C5⋊S4)⋊C2
unambiguous semidirect product(D5×A4)⋊C2

Splittings

Fitting, socleC22×C5 ⋊ C2×D3
derivedC5×A4 ⋊ C22
SylowC5 ⋊ C2×S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC5×A4C22
Fitting, socleC22×C5C2×D3
LCSG ▷ C5×A4
UCSC1

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,4,3)(5,6,8,9,7);(2,3,4)(6,7)(8,9)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×S4, C2×D5×S4, C3×D5×S4, …
maximal normal subgroup ofAGL(1,𝔽5)×S4, C2×D5×S4, C3×D5×S4, …
maximal quotient of#91891, #91892, #91893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 194);