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Cayley diagram

C3×#3777 #88944

order432
isoclinismisoclinic class of PSU(3,𝔽2)
automorphismAut(G) = C2×#10596418 (order 1152) · Inn(G) = PSU(3,𝔽2) · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×#3777

Splittings

SylowC33 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C3PSU(3,𝔽2)
commutatorC3⋊D3C2×C3×C4
FrattiniC2C3×PSU(3,𝔽2)
Fitting, socleC2×C33Q8
prime radicalC2×C3×C3⋊D3C22
cosocleC2×C3⋊D3C22×C3
hypocenterC32C3×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×GO+(4,𝔽2)⋊C4, C3×AGL(1,𝔽9)⋊C4, #11724101, …
maximal normal subgroup ofC3×GO+(4,𝔽2)⋊C4, C3×AGL(1,𝔽9)⋊C4, #11724101, …
maximal quotient ofC3×#63744, C3×#63745, C3×#63746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 591);