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H(3,𝔽₃)⋊(C₃₁⋊C₉) #21419234

order	7533
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₃²

Aliases

unambiguous semidirect product

H(3,𝔽₃)⋊(C₃₁⋊C₉)

Splittings

C₃₁ ⋊ H(3,𝔽₃)⋊C₉

Subgroups

	subgroup	G/H
center	C₃²	C₃²⋊(C₃₁⋊C₃)
commutator	C₃²×C₃₁	C₃×C₉
Frattini	C₃³	C₃×C₃₁⋊C₃
Fitting	C₃×H(3,𝔽₃)×C₃₁	C₃
socle	C₃²×C₃₁	H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

3^5,31|a^3=d,b^a=bc,c^a=ce,f^a=f^25,c^b=ce

Conjugacy classes & character table

381 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁⋊C₃×H(3,𝔽₃)⋊C₉, C₃×H(3,𝔽₃)⋊(C₃₁⋊C₉), #36381019, …
maximal normal subgroup of	C₃₁⋊C₃×H(3,𝔽₃)⋊C₉, C₃×H(3,𝔽₃)⋊(C₃₁⋊C₉), #36381019, …
maximal quotient of	H(3,𝔽₃)⋊(C₃₁⋊C₂₇), #36379503, #36379505, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7533, 104);