C₂×A₄×C₁₉ #90729

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

Aliases

direct product	C₂×A₄×C₁₉
unambiguous semidirect product	C₂³⋊(C₃×C₁₉)

	subgroup	G/H
center, hypercenter	C₂×C₁₉	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₂³×C₁₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₁₉

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 114

A minimal faithful permutation representation — 2 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 49);