C₂²×#126⋊C₃ #85589

order	384
isoclinism	isoclinic class of #126⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

Fitting, Sylow	C₂²×#126 ⋊ C₃
derived	C₄² ⋊ C₂³×C₃

	subgroup	G/H
center, hypercenter	C₂²	#126⋊C₃
commutator, cosocle, hypocenter	C₄²	C₂³×C₃
Frattini	C₂²	C₂³×A₄
Fitting	C₂²×#126	C₃
socle	C₂⁴	C₂×A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₂²

300 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18165);