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C₃×#4008 #91212

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×D₄ ⋊ C₄
Hall	C₃×C₅ ⋊ #104
Sylow	C₅ ⋊ C₃×#104

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×D₄	C₄
socle	C₂×C₃×C₅	C₂²⋊C₄
prime radical	C₃×#711	C₂
cosocle	C₄×D₅	C₂²×C₃
hypercenter	C₃×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#104
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₄

88 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)(6,8)(7,10)(9,12)(11,13)(14,15,16);(2,5,4,3)(6,8,10,7,11,13,9,12)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11751935, D₃×#4008, #11751949, …
maximal normal subgroup of	#11751935, D₃×#4008, #11751949, …
maximal quotient of	C₃×#64869, C₃×#64870, C₃×#64872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 288);