C₂×C₃×#124 #5856

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

330 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1415);