order	96
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = C₂×#442 (order 128) · Inn(G) = D₈ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

general semilinear	ΓL(1,𝔽₄₉)
direct product	C₃×SD₃₂

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	D₈
commutator, Frattini, cosocle	C₈	C₂²×C₃
prime radical	C₃×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ G

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,5,9)(3,6,12,15)(4,13,11,8)(7,16,14,10)(17,18,19);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 62);