C₂×C₃×#544 #87341

order	384
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

1080 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19917);