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C₂×C₅×C₂²⋊C₉ #67116

order	360
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂×C₅×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×C₂²⋊C₉
unambiguous semidirect product	C₂³⋊(C₅×C₉)
unambiguous semidirect product	(C₂³×C₅)⋊C₉

Splittings

derived	C₂² ⋊ C₂×C₅×C₉
Hall	C₂³×C₅ ⋊ C₉
Sylow	C₂³ ⋊ C₅×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	A₄
commutator, hypocenter	C₂²	C₂×C₅×C₉
Frattini	C₃	C₂×C₅×A₄
Fitting, socle, prime radical	C₂³×C₃×C₅	C₃
cosocle	C₂²×C₃	C₂×C₃×C₅
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₃×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(4,5,6)(7,8,9,10,11)(12,16,20,14,19,15,17,13,18);(3,6)(4,5)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13085154, #13085157, #13085159, …
maximal subgroup of	C₅×(C₂²⋊C₉)⋊C₄, (C₂²⋊C₉)⋊Dic₅, Dic₅×C₂²⋊C₉, …
maximal normal subgroup of	C₅×(C₂²⋊C₉)⋊C₄, (C₂²⋊C₉)⋊Dic₅, Dic₅×C₂²⋊C₉, …
maximal quotient of	C₄×C₅×C₂²⋊C₉, C₂×C₅×Q₈⋊C₉, C₅×#41⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 46);