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C₅×C₉ #194

order	45
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₄ (order 24) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₅
abelian	C₅×C₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃×C₅
socle, prime radical	C₃×C₅	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 1 generator on 14 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

center of	C₅×D₃×C₉, C₅×C₂₇⋊C₃, C₅×#737, …
derived subgroup of	D₄₅, Dic₄₅, C₉⋊AGL(1,𝔽₅), …
Frattini subgroup of	C₂₅×C₂₇, C₂₅×D₂₇, C₂₇×D₂₅, …
Fitting subgroup of	C₅×D₉, C₉×D₅, D₄₅, …
maximal subgroup of	C₅×D₉, C₉×D₅, D₄₅, …
maximal normal subgroup of	C₅×D₉, C₉×D₅, D₄₅, …
maximal quotient of	C₂×C₅×C₉, C₅×C₂₇, C₃×C₅×C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(45, 1);