C₂×C₃²×AGL(2,𝔽₃) #72228425

order	7776
isoclinism	isoclinic class of AGL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting, socle	C₂×C₃⁴ ⋊ GL(2,𝔽₃)
derived	PSU(3,𝔽₂)⋊C₃ ⋊ C₂²×C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃²	AGL(2,𝔽₃)
commutator, cosocle, hypocenter	PSU(3,𝔽₂)⋊C₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃⁴	GL(2,𝔽₃)
LCS	G ▷ PSU(3,𝔽₂)⋊C₃
UCS	C₁ ◁ C₂×C₃²

1354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 17 points:

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

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,5,6)(7,12,14,10,9,8)(11,13)(16,17), (1,2,3)(8,9)(10,15)(11,14));