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Cayley diagram

C7×PSU(3,𝔽2) #92744

order504
isoclinismisoclinic class of PSU(3,𝔽2)
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×PSU(3,𝔽2)

Splittings

Fitting, socle, HallC32×C7 ⋊ Q8
SylowC32 ⋊ C7×Q8

Subgroups

subgroupG/H
center, hypercenterC7PSU(3,𝔽2)
commutator, cosocleC3⋊D3C22×C7
FrattiniC1G
Fitting, socleC32×C7Q8
prime radicalC7×C3⋊D3C22
hypocenterC32C7×Q8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C7

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

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

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7⋊(PSU(3,𝔽2)⋊C3), C2×C7⋊(PSU(3,𝔽2)⋊C3), C2×ΓL(1,𝔽8)×PSU(3,𝔽2)⋊C3, …
maximal subgroup ofC7×AΓL(1,𝔽9), PSU(3,𝔽2)⋊D7, D7×PSU(3,𝔽2), …
maximal normal subgroup ofC7×AΓL(1,𝔽9), PSU(3,𝔽2)⋊D7, D7×PSU(3,𝔽2), …
maximal quotient ofC7×#3777, C2×C7×PSU(3,𝔽2), C7×SU(3,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 170);