C₃×D₃×SL(2,𝔽₃) #88976

order	432
isoclinism	isoclinic class of C₃⋊(#41⋊C₃)
automorphism	Aut(G) = C₂×D₃²×S₄ (order 1728) · Inn(G) = D₃×A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×D₃×SL(2,𝔽₃)
unambiguous semidirect product	C₃⋊(C₂×C₃×SL(2,𝔽₃))
unambiguous semidirect product	(D₃×Q₈)⋊C₃²
unambiguous semidirect product	(C₃×D₃×Q₈)⋊C₃

Fitting	C₃²×Q₈ ⋊ C₂×C₃
derived	C₃×Q₈ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂×C₃²
Frattini	C₂	C₃×D₃×A₄
Fitting	C₃²×Q₈	C₂×C₃
socle	C₂×C₃²	C₂×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂×C₃

125 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 623);