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C₃×#3992 #91196

order	480
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃×C₅×C₈ ⋊ C₄
Hall	C₃×C₅ ⋊ #97
Sylow	C₅ ⋊ C₃×#97

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃×C₄²
Frattini	C₄	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₈	C₄
socle	C₂×C₃×C₅	C₄²
prime radical	C₂×C₃×D₅	C₂×C₄
cosocle	C₄×D₅	C₂²×C₃
hypercenter	C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₃×C₈

80 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)(6,11,12,7,9,13,8,10)(14,15,16);(2,5,4,3)(7,10)(11,13)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746452, #11746453, C₃×#64845, …
maximal normal subgroup of	#11746452, #11746453, C₃×#64845, …
maximal quotient of	C₃×#64845, C₃×#64846, C₃×#64879, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 272);