C₃×#550 #5880

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 6144
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

312 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1439);