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Cayley diagram

C4×S4 #979

order96
isoclinismisoclinic class of S4
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = S4 · Out(G) = C22
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×S4

Splittings

FittingC22×C4 ⋊ D3
derivedA4 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4S4
commutator, hypocenterA4C2×C4
FrattiniC2C2×S4
FittingC22×C4D3
socleC23C2×D3
prime radicalC2×S4C2
cosocleC2×A4C22
LCSG ▷ A4
UCSC1 ◁ C4

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×S4, #5400, C2×C4×S4, …
maximal normal subgroup ofC8×S4, #5400, C2×C4×S4, …
maximal quotient ofC4×#226, #5388, C4×GL(2,𝔽3), …
automorphism group ofC5×Q8, C5×A4, C5×SL(2,𝔽3), …
outer automorphism group ofC5×C3⋊D3, C22×C5×D3, C22×D15, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 186);