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C₃²×#239 #89060

order	432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₃²×#239

Splittings

Fitting	C₃³×Q₈ ⋊ C₂
Sylow	C₃³ ⋊ #41

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₃²
Frattini	C₂	C₂²×C₃²×D₃
Fitting	C₃³×Q₈	C₂
socle	C₂×C₃³	C₂³
hypercenter	C₃²×Q₈	D₃
hypocenter	C₃	C₃²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×Q₈

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

(4,5,6)(7,8,9)(10,16,13,12)(11,17,15,14);(1,2,3)(10,15,13,11)(12,14,16,17);(8,9)(12,16)(14,17)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#63939, C₃×#63943, C₃×#63947, …
maximal normal subgroup of	C₃×#63939, C₃×#63943, C₃×#63947, …
maximal quotient of	C₃²×#890, C₃²×#892, C₃²×#893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 707);