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Cayley diagram

C2×C3×PSU(3,𝔽2) #89110

order432
isoclinismisoclinic class of PSU(3,𝔽2)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3

Aliases

direct productC2×C3×PSU(3,𝔽2)

Splittings

Fitting, socleC2×C33 ⋊ Q8
SylowC33 ⋊ C2×Q8

Subgroups

subgroupG/H
center, hypercenterC2×C3PSU(3,𝔽2)
commutator, cosocleC3⋊D3C23×C3
FrattiniC1G
Fitting, socleC2×C33Q8
prime radicalC2×C3×C3⋊D3C22
hypocenterC32C2×C3×Q8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of#67×PSU(3,𝔽2)⋊C3, C3×#67×PSU(3,𝔽2)⋊C3, D12×PSU(3,𝔽2)⋊C3
maximal subgroup ofC3×PSU(3,𝔽2)⋊C4, #11724102, Dic3×PSU(3,𝔽2), …
maximal normal subgroup ofC3×PSU(3,𝔽2)⋊C4, #11724102, Dic3×PSU(3,𝔽2), …
maximal quotient ofC3×#64243, C3×C4×PSU(3,𝔽2), C3×C4⋊PSU(3,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 757);