Finite Group Databasebeta

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C₃×D₃×D₅×AGL(1,𝔽₅) #72231111

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₃×D₃×D₅×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃²×C₅² ⋊ C₂²×C₄
derived	C₃×C₅² ⋊ C₂²×C₃×C₄
Sylow	C₅² ⋊ C₂×C₃×C₄×D₃
Sylow	C₃² ⋊ C₂×D₅×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₃×D₅×AGL(1,𝔽₅)
commutator, hypocenter	C₃×C₅²	C₂²×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅²	C₂²×C₄
prime radical	C₃×D₃×D₅²	C₂
cosocle	C₃×C₅×D₅	C₂³×C₃
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₃

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×AGL(1,𝔽₅)², D₃²×D₅×AGL(1,𝔽₅), D₃×AGL(1,𝔽₅)×ΓL(1,𝔽₁₆), …
maximal normal subgroup of	C₃×D₃×AGL(1,𝔽₅)², D₃²×D₅×AGL(1,𝔽₅), D₃×AGL(1,𝔽₅)×ΓL(1,𝔽₁₆)
maximal quotient of	D₅×AGL(1,𝔽₅)×AGL(1,ℤ/9)

GAP

Past the SmallGroups library coverage — no SmallGroups id.