3₋¹⁺²×#458 #13145134

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³

Aliases

direct product

3₋¹⁺²×#458

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#120
commutator, Frattini, cosocle	C₂³×C₃	C₂³×C₃²
prime radical	C₃²×H(5,𝔽₂)	C₂×C₃
LCS	G ▷ C₂³×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×3₋¹⁺² ◁ G

640 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,4,3,6,8,7,5,9)(10,11,14,17)(12,15,19,22)(13,16,20,23)(18,24,25,21);(2,5,6)(4,8,9)(10,12)(11,15)(13,18)(14,19)(16,21)(17,22)(20,25)(23,24);(10,13)(11,16)(12,18)(14,20)(15,21)(17,23)(19,25)(22,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11197);