H(3,𝔽₃)×#410 #13144989

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃²×C₄

Aliases

direct product	H(3,𝔽₃)×#410
unambiguous semidirect product	(C₃²×#410)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃²×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₃²×C₄
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂×C₃	C₂×C₃²×C₂²⋊C₄
prime radical	C₂×#41×H(3,𝔽₃)	C₂
LCS	G ▷ C₂²×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₄×H(3,𝔽₃) ◁ G

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

normal subgroups only

This group has 869 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)(12,18,20,15)(13,16)(14,17)(19,24,25,22)(21,23);(2,5,6)(4,8,9)(10,12)(11,15)(13,19)(14,20)(16,22)(17,18)(21,25)(23,24);(10,13,14,21)(11,16,17,23)(12,19,20,25)(15,22,18,24)

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, 11052);