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C₅×C₂²⋊C₉ #4363

order	180
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₅×C₉ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

derived, Sylow	C₂² ⋊ C₅×C₉
Hall	C₂²×C₅ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	A₄
commutator, hypocenter	C₂²	C₅×C₉
Frattini	C₃	C₅×A₄
Fitting, socle, prime radical	C₂²×C₃×C₅	C₃
cosocle	C₂²×C₃	C₃×C₅
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3,4)(5,6,7,8,9)(10,11,13,12,14,16,15,17,18);(1,4)(2,3)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₅⋊((C₂²⋊C₉)⋊C₂), (C₂²⋊C₉)⋊Dic₅, (C₂²⋊C₉)⋊AGL(1,𝔽₅), …
maximal subgroup of	C₅×(C₂²⋊C₉)⋊C₂, C₅⋊((C₂²⋊C₉)⋊C₂), D₅×C₂²⋊C₉, …
maximal normal subgroup of	C₅×(C₂²⋊C₉)⋊C₂, C₅⋊((C₂²⋊C₉)⋊C₂), D₅×C₂²⋊C₉, …
maximal quotient of	C₅×Q₈⋊C₉, C₂×C₅×C₂²⋊C₉, C₅×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 8);