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C₅×PSU(3,𝔽₂) #67205

order	360
isoclinism	isoclinic class of PSU(3,𝔽₂)
automorphism	Aut(G) = C₄×AGL(2,𝔽₃) (order 1728) · Inn(G) = PSU(3,𝔽₂) · Out(G) = C₄×D₃
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₃

Aliases

C₅×PSU(3,𝔽₂)

Splittings

Fitting, socle, Hall	C₃²×C₅ ⋊ Q₈
Sylow	C₃² ⋊ C₅×Q₈

Subgroups

	subgroup	G/H
center, hypercenter	C₅	PSU(3,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅	Q₈
prime radical	C₅×C₃⋊D₃	C₂²
hypocenter	C₃²	C₅×Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₅

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×AΓL(1,𝔽₉), PSU(3,𝔽₂)⋊D₅, D₅×PSU(3,𝔽₂), …
maximal normal subgroup of	C₅×AΓL(1,𝔽₉), PSU(3,𝔽₂)⋊D₅, D₅×PSU(3,𝔽₂), …
maximal quotient of	C₅×#3777, C₂×C₅×PSU(3,𝔽₂), C₅×SU(3,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 135);