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C₃×ΓL(1,𝔽₁₆) #4376

order	180
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₅) (order 240) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×ΓL(1,𝔽₁₆)
unambiguous semidirect product	C₃⋊(C₃×AGL(1,𝔽₅))

Splittings

Fitting, socle, Hall	C₃²×C₅ ⋊ C₄
derived	C₃×C₅ ⋊ C₃×C₄
Sylow	C₃² ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₃×Dic₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₄
prime radical	C₃²×D₅	C₂
cosocle	C₃×D₅	C₂×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,3)(5,6)(8,11,10,9);(4,5,6)(7,8,11,9,10)

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×AGL(1,𝔽₅), D₃×ΓL(1,𝔽₁₆), ΓL(1,𝔽₁₆)⋊D₃, …
maximal normal subgroup of	C₃×D₃×AGL(1,𝔽₅), D₃×ΓL(1,𝔽₁₆), ΓL(1,𝔽₁₆)⋊D₃, …
maximal quotient of	C₃×C₃⋊#170, C₂×C₃×ΓL(1,𝔽₁₆), C₃×C₉⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 21);