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C₂×ΓL(1,𝔽₁₆) #1231

order	120
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) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×ΓL(1,𝔽₁₆)
unambiguous semidirect product	(C₂×C₃)⋊AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊Dic₃

Splittings

Fitting, socle	C₂×C₃×C₅ ⋊ C₄
derived, Hall	C₃×C₅ ⋊ C₂×C₄
Sylow	C₅ ⋊ C₂×Dic₃
Sylow	C₃ ⋊ C₂×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅	C₄
prime radical	C₂×C₃×D₅	C₂
cosocle	C₃×D₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×AGL(1,𝔽₅), (C₂×D₃)⋊AGL(1,𝔽₅), Dic₃⋊AGL(1,𝔽₅), …
maximal normal subgroup of	Dic₃×AGL(1,𝔽₅), (C₂×D₃)⋊AGL(1,𝔽₅), Dic₃⋊AGL(1,𝔽₅), …
maximal quotient of	D₅⋊(C₃⋊C₈), #6758, C₄×ΓL(1,𝔽₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 41);