C₂²×#1020 #87587

order	384
isoclinism	isoclinic class of #1020
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

Fitting, socle	C₂⁶ ⋊ D₃
derived	C₂²⋊A₄ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	#1020
commutator, cosocle, hypocenter	C₂²⋊A₄	C₂³
Frattini	C₁	G
Fitting, socle	C₂⁶	D₃
LCS	G ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂²

1427 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 3 generators on 12 points:

(1,3)(2,5)(4,11,10,9)(6,12,7,8);(1,7,9,3,4,12)(2,10,8,5,6,11);(1,2)(3,5)(4,6)(7,10)(8,9)(11,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20163);