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C₃×#3998 #91202

order	480
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₂×#41×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₅ ⋊ M₃₂
Sylow	C₅ ⋊ C₃×M₃₂

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₈
Frattini	C₄	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₄×C₅	C₄
socle	C₂×C₃×C₅	C₂×C₈
prime radical	C₂²×C₃×C₅	C₈
cosocle	#168	C₂²×C₃
hypercenter	C₂×C₃×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,5,4)(7,11)(12,17)(13,18)(19,21)(22,23,24);(2,5,4,3)(6,12,14,11,16,21,8,13,10,17,20,7,9,19,15,18)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746544, #11746545, #11746548, …
maximal normal subgroup of	#11746544, #11746545, #11746548, …
maximal quotient of	C₃×C₄⋊#674, C₃×#64859, C₃×C₂²⋊#674, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 278);