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C₅²×C₂²⋊C₉ #11744653

order	900
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₅²×C₉ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅²×C₂²⋊C₉
unambiguous semidirect product	C₂²⋊(C₅²×C₉)
unambiguous semidirect product	(C₂²×C₅)⋊(C₅×C₉)

Splittings

derived, Sylow	C₂² ⋊ C₅²×C₉
Hall	C₂²×C₅² ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅²	A₄
commutator, hypocenter	C₂²	C₅²×C₉
Frattini	C₃	C₅²×A₄
Fitting, socle, prime radical	C₂²×C₃×C₅²	C₃
cosocle	C₂²×C₃	C₃×C₅²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅²

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 450

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	C₅⋊(C₅⋊((C₂²⋊C₉)⋊C₂))
maximal subgroup of	C₅²×(C₂²⋊C₉)⋊C₂, C₅×C₅⋊((C₂²⋊C₉)⋊C₂), C₅⋊(C₅⋊((C₂²⋊C₉)⋊C₂)), …
maximal normal subgroup of	C₅²×(C₂²⋊C₉)⋊C₂, C₅×C₅⋊((C₂²⋊C₉)⋊C₂), C₅⋊(C₅⋊((C₂²⋊C₉)⋊C₂)), …
maximal quotient of	C₅²×Q₈⋊C₉, C₂×C₅²×C₂²⋊C₉, C₅²×C₁₃×C₂²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(900, 65);