H(3,𝔽₃)×#381 #13144902

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃²

Aliases

direct product	H(3,𝔽₃)×#381
unambiguous semidirect product	(C₃²×#381)⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂⁴×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

1199 conjugacy classes of subgroups; only the 343 normal subgroups are shown.

normal subgroups only

This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 343 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) = 288

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

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10965);