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Cayley diagram

A4×C32 #68043

order384
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C8×S4 (order 384) · Inn(G) = A4 · Out(G) = C22×C8
homologyAbelianization (H1) = C3×C32 · Schur multiplier (H2) = C2

Aliases

direct productA4×C32
unambiguous semidirect product(C22×C32)⋊C3

Splittings

Fitting, SylowC22×C32 ⋊ C3
derivedC22 ⋊ C3×C32

Subgroups

subgroupG/H
center, hypercenterC32A4
commutator, hypocenterC22C3×C32
FrattiniC16C2×A4
FittingC22×C32C3
socleC23C3×C16
prime radicalC2×A4C16
cosocleC22×C16C2×C3
LCSG ▷ C22
UCSC1 ◁ C32

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 36 points:

(2,3,4)(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36);(1,4)(2,3)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×C64, C2×A4×C32, A4×M64, …
maximal normal subgroup ofA4×C64, C2×A4×C32, A4×M64, …
maximal quotient ofSL(2,𝔽3)×C32, A4×C64, #13737⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 619);