H(3,𝔽₃)×#430 #13145049

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#430
unambiguous semidirect product	(C₃²×#430)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×C₃²×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂×C₃	C₂×C₃²×C₄⋊C₄
prime radical	C₂³×H(3,𝔽₃)	Q₈
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₄×H(3,𝔽₃) ◁ G

561 conjugacy classes of subgroups; only the 273 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 273 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)(5,9,7)(10,11,14,17,15,18,13,23)(12,19,22,24,20,25,21,16);(2,5,6)(4,8,9)(10,12,13,21,15,20,14,22)(11,16,17,19,18,24,23,25);(12,20)(16,24)(19,25)(21,22)

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11112);