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#135×(C₃⋊D₃)⋊C₃ #13153481

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product	#135×(C₃⋊D₃)⋊C₃
unambiguous semidirect product	(C₃⋊D₃×#135)⋊C₃

Splittings

Fitting	H(3,𝔽₃)×#135 ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ C₂×#135

Subgroups

	subgroup	G/H
center	C₄	D₄×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	H(3,𝔽₃)×#135	C₂
socle	C₂×C₃	C₂×C₃×D₃×D₄
hypercenter	#135	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 144

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#135×PSU(3,𝔽₂)⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19544);