Finite Group Databasebeta

← Prev (#73032) Next (#73034) →

Cayley diagram

C16×S4 #73033

order384
isoclinismisoclinic class of S4
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = S4 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C16 · Schur multiplier (H2) = C22

Aliases

direct productC16×S4

Splittings

FittingC22×C16 ⋊ D3
derivedA4 ⋊ C2×C16

Subgroups

subgroupG/H
center, hypercenterC16S4
commutator, hypocenterA4C2×C16
FrattiniC8C2×S4
FittingC22×C16D3
socleC23D3×C8
prime radicalC2×S4C8
cosocleC8×A4C22
LCSG ▷ A4
UCSC1 ◁ C16

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,4,3);(2,3,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×C32, #11711659, C2×C16×S4, …
maximal normal subgroup ofS4×C32, #11711659, C2×C16×S4, …
maximal quotient ofC16×GL(2,𝔽3), C16×#226, GL(2,𝔽3)⋊C16, …
automorphism group ofQ8×C17, A4×C17, C17×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5609);