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C₃×AGL(1,𝔽₁₃) #90854

order	468
isoclinism	isoclinic class of AGL(1,𝔽₁₃)
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = AGL(1,𝔽₁₃) · Out(G) = D₃
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×AGL(1,𝔽₁₃)
unambiguous semidirect product	#261⋊C₃²
unambiguous semidirect product	(C₃×#261)⋊C₃

Splittings

Fitting, socle	C₃×C₁₃ ⋊ C₃×C₄
derived, Sylow	C₁₃ ⋊ C₃²×C₄
Hall	C₃×C₁₃⋊C₃ ⋊ C₄
Hall	#261 ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₁₃	C₃×C₄
prime radical	C₃×D₁₃⋊C₃	C₂
cosocle	D₁₃	C₂×C₃²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 39

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

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₁₃), C₂×C₃×AGL(1,𝔽₁₃), C₉×AGL(1,𝔽₁₃), …
maximal normal subgroup of	D₃×AGL(1,𝔽₁₃), C₂×C₃×AGL(1,𝔽₁₃), C₉×AGL(1,𝔽₁₃), …
maximal quotient of	C₃×#1057⋊C₃, C₂×C₃×AGL(1,𝔽₁₃), C₉×AGL(1,𝔽₁₃), …
automorphism group of	C₁₃⋊C₉, D₁₃⋊C₉, C₂×C₁₃⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 29);