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C₂×C₉×GL(2,𝔽₃) #11722326

order	864
isoclinism	isoclinic class of #226
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

solvable

Aliases

C₂×C₉×GL(2,𝔽₃)

Splittings

Fitting	C₂×Q₈×C₉ ⋊ D₃
derived	SL(2,𝔽₃) ⋊ C₂²×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₉	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₉
Frattini	C₂×C₃	C₂×C₃×S₄
Fitting	C₂×Q₈×C₉	D₃
socle	C₂²×C₃	C₃×S₄
prime radical	C₂×C₃×GL(2,𝔽₃)	C₃
cosocle	C₃×SL(2,𝔽₃)	C₂²×C₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₉

167 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 144

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13146211, #13146212, #13146213, …
maximal normal subgroup of	#13146211, #13146212, #13146213, …
maximal quotient of	C₉×#5386, C₄×C₉×GL(2,𝔽₃), C₉×C₄⋊GL(2,𝔽₃), …
automorphism group of	C₃²×C₁₉, C₂×C₃²×C₁₉
outer automorphism group of	C₃²×C₁₉, C₂×C₃²×C₁₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 2293);