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C₃×C₉×SL(2,𝔽₃) #10619910

order	648
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉ · Schur multiplier (H₂) = C₃³

solvable

Aliases

C₃×C₉×SL(2,𝔽₃)

Splittings

Fitting	C₃×Q₈×C₉ ⋊ C₃
derived, Sylow	Q₈ ⋊ C₃²×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₉	A₄
commutator, hypocenter	Q₈	C₃²×C₉
Frattini	C₂×C₃	C₃²×A₄
Fitting	C₃×Q₈×C₉	C₃
socle	C₂×C₃²	C₃×A₄
prime radical	C₃²×SL(2,𝔽₃)	C₃
cosocle	C₃×Q₈	C₃³
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₉

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 216

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊(C₉⋊#226), C₃⋊(C₉⋊GL(2,𝔽₃))
maximal subgroup of	C₃×C₉×#226, C₃×C₉×GL(2,𝔽₃), C₉×C₃⋊#226, …
maximal normal subgroup of	C₃×C₉×#226, C₃×C₉×GL(2,𝔽₃), C₉×C₃⋊#226, …
maximal quotient of	C₂×C₃×C₉×SL(2,𝔽₃), C₉²×SL(2,𝔽₃), #14517166, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 429);