C₃×#559 #5889

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) = C₂×#72215688 (order 4096) · Inn(G) = C₂⁴ · Out(G) = C₂×#2196
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₃×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

236 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1448);