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C₃×#4009 #91213

order	480
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂²⋊AGL(1,𝔽₅) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₄
Hall	C₃×C₅ ⋊ #103
Sylow	C₅ ⋊ C₃×#103

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄
Frattini	C₄	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×Q₈	C₄
socle	C₂×C₃×C₅	C₂²⋊C₄
prime radical	C₂×C₃×D₅	D₄
cosocle	C₄×D₅	C₂²×C₃
hypercenter	C₃×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#103
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4009, #11751961, #11751962, …
maximal normal subgroup of	D₃×#4009, #11751961, #11751962, …
maximal quotient of	C₃×#64869, C₃×#64872, C₃×#64873, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 289);