C₂×C₃×#125 #5857

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

Aliases

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

226 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,4,5);(8,12)(10,13)(14,15,17,18)(16,19,20,21);(6,7,9,11)(8,10,12,13)(15,18)(19,21);(1,2)(6,8)(7,10)(9,12)(11,13)(14,16,17,20)(15,19,18,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1416);