Finite Group Databasebeta

← Prev (#4643) Next (#4645) →

Cayley diagram

A4×C16 #4644

order192
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = A4 · Out(G) = C22×C4
homologyAbelianization (H1) = C3×C16 · Schur multiplier (H2) = C2

Aliases

direct productA4×C16
unambiguous semidirect product(C22×C16)⋊C3

Splittings

Fitting, SylowC22×C16 ⋊ C3
derivedC22 ⋊ C3×C16

Subgroups

subgroupG/H
center, hypercenterC16A4
commutator, hypocenterC22C3×C16
FrattiniC8C2×A4
FittingC22×C16C3
socleC23C3×C8
prime radicalC2×A4C8
cosocleC22×C8C2×C3
LCSG ▷ C22
UCSC1 ◁ C16

33 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:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711662, #11711663, #11711664, …
maximal subgroup ofA4⋊C32, A4×C32, C16×S4, …
maximal normal subgroup ofA4⋊C32, A4×C32, C16×S4, …
maximal quotient ofC16×SL(2,𝔽3), A4×C32, #2258⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 203);