order	456
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂×C₉×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₃×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊(C₃×C₁₉)

Splittings

Fitting, Hall	Q₈×C₁₉ ⋊ C₃
derived, Sylow	Q₈ ⋊ C₃×C₁₉

	subgroup	G/H
center, socle, hypercenter	C₂×C₁₉	A₄
commutator, cosocle, hypocenter	Q₈	C₃×C₁₉
Frattini	C₂	A₄×C₁₉
Fitting	Q₈×C₁₉	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₁₉

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 152

A minimal faithful permutation representation — 2 generators on 27 points:

(2,3,5)(6,8,7)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(1,5,4,7)(2,8,6,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 22);