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C4×D4×S4 #11714307

order768
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC4×D4×S4

Splittings

FittingC22×C4×D4 ⋊ D3

Subgroups

subgroupG/H
centerC2×C4C22×S4
commutatorC2×A4C23×C4
FrattiniC22C23×S4
FittingC22×C4×D4D3
socleC24C23×D3
prime radicalC2×D4×S4C2
cosocleC22×A4C24
hypercenterC4×D4S4
hypocenterA4C2×C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4×D4

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 48

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×S42, #72228056, D42×S4, …
maximal normal subgroup of#72228056, D42×S4, S4×#442, …
maximal quotient ofD4×AGL(1,𝔽5)×S4, S4×#442, S4×#444, …
outer automorphism group ofC2×C5×#41

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1088688);