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C₃×PSU(3,𝔽₂) #6323

order	216
isoclinism	isoclinic class of PSU(3,𝔽₂)
automorphism	Aut(G) = C₂×AGL(2,𝔽₃) (order 864) · Inn(G) = PSU(3,𝔽₂) · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₃

Aliases

C₃×PSU(3,𝔽₂)

Splittings

Fitting, socle, Sylow

C₃³ ⋊ Q₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃	PSU(3,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃³	Q₈
prime radical	C₃×C₃⋊D₃	C₂²
hypocenter	C₃²	C₃×Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₃

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

derived subgroup of	D₃×PSU(3,𝔽₂)⋊C₃, C₉⋊(PSU(3,𝔽₂)⋊C₃), #14519057, …
maximal subgroup of	C₃×AΓL(1,𝔽₉), #89092, D₃×PSU(3,𝔽₂), …
maximal normal subgroup of	C₃×AΓL(1,𝔽₉), #89092, D₃×PSU(3,𝔽₂), …
maximal quotient of	C₃×#3777, C₂×C₃×PSU(3,𝔽₂), C₉×PSU(3,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 160);