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C₂²×ΓL(1,𝔽₃₂)⋊C₉ #19150033

order	5580
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×ΓL(1,𝔽₃₂)⋊C₉
unambiguous semidirect product	(C₂²×C₃₁⋊C₉)⋊C₅

Splittings

derived, Sylow	C₃₁ ⋊ C₂²×C₅×C₉
Hall	C₂²×C₃₁⋊C₉ ⋊ C₅
Hall	C₂²×ΓL(1,𝔽₃₂) ⋊ C₉
Hall	C₃₁⋊C₉ ⋊ C₂²×C₅
Hall	ΓL(1,𝔽₃₂) ⋊ C₂²×C₉
Hall	C₂²×C₃₁ ⋊ C₅×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃	(C₃₁⋊C₃)⋊C₅
commutator, hypocenter	C₃₁	C₂²×C₅×C₉
Frattini	C₃	C₂²×(C₃₁⋊C₃)⋊C₅
Fitting, socle	C₂²×C₃×C₃₁	C₃×C₅
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂²×C₃

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₅×ΓL(1,𝔽₃₂)⋊C₉, C₂²×C₅×ΓL(1,𝔽₃₂)⋊C₉
maximal normal subgroup of	C₂²×C₅×ΓL(1,𝔽₃₂)⋊C₉
maximal quotient of	C₂²×(C₃₁⋊C₉)⋊C₂₅, C₂×D₅×ΓL(1,𝔽₃₂)⋊C₉, C₂²×C₅×ΓL(1,𝔽₃₂)⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5580, 22);