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C₃×C₉×D₅ #63187

order	270
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₉×D₅
unambiguous semidirect product	C₅⋊(C₂×C₃×C₉)

Splittings

Fitting, Hall	C₃×C₅×C₉ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₉	D₅
commutator, hypocenter	C₅	C₂×C₃×C₉
Frattini	C₃	C₃²×D₅
Fitting	C₃×C₅×C₉	C₂
socle	C₃²×C₅	C₂×C₃
prime radical	C₃²×D₅	C₃
cosocle	C₃×C₅	C₂×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 135

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×C₉×AGL(1,𝔽₅), C₃×C₉⋊AGL(1,𝔽₅), C₉×ΓL(1,𝔽₁₆), …
maximal normal subgroup of	C₃×C₉×AGL(1,𝔽₅), C₃×C₉⋊AGL(1,𝔽₅), C₉×ΓL(1,𝔽₁₆), …
maximal quotient of	C₃×C₉×Dic₅, C₂×C₃×C₉×D₅, C₉²×D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 5);