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Cayley diagram

C23×C3×A4 #64393

order288
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC23×C3×A4

Splittings

Fitting, socleC25×C3 ⋊ C3
derivedC22 ⋊ C23×C32
SylowC25 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC23×C3A4
commutator, cosocle, hypocenterC22C23×C32
FrattiniC1G
Fitting, socleC25×C3C3
LCSG ▷ C22
UCSC1 ◁ C23×C3

420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 13 points:

(5,6)(11,12,13);(3,4)(7,8,9)(10,13)(11,12);(1,2)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×#64209, A4×#64269, A4×#64210, …
maximal subgroup ofC3×#5432, #10593666, A4×#217, …
maximal normal subgroup ofC3×#5432, #10593666, A4×#217, …
maximal quotient ofC22×C3×C4×A4, C2×C3×D4×A4, C23×C3×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1041);