C₃×#546 #5876

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) = C₂⁵×#120 (order 1024) · Inn(G) = C₂⁴ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

334 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1435);