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Cayley diagram

C4×C9×A4 #88678

order432
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = A4 · Out(G) = C22×C3×D3
homologyAbelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C2×C3

Aliases

direct productC4×C9×A4

Splittings

FittingC22×C4×C9 ⋊ C3
derivedC22 ⋊ C3×C4×C9
SylowC22×C4 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC4×C9A4
commutator, hypocenterC22C3×C4×C9
FrattiniC2×C3C2×C3×A4
FittingC22×C4×C9C3
socle, cosocleC23×C3C2×C32
prime radicalC2×C3×A4C2×C3
LCSG ▷ C22
UCSC1 ◁ C4×C9

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146193, #13146194, #13146195, …
maximal subgroup ofC9×A4⋊C8, C9⋊(A4⋊C8), A4×C9⋊C8, …
maximal normal subgroup ofC9×A4⋊C8, C9⋊(A4⋊C8), A4×C9⋊C8, …
maximal quotient ofC4×C9×SL(2,𝔽3), C8×C9×A4, C9×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 325);