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C₂²×ΓL(1,𝔽₁₆) #6840

order	240
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = D₃×AGL(1,𝔽₅)×S₄ (order 2880) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×ΓL(1,𝔽₁₆)
unambiguous semidirect product	D₅⋊(C₂×Dic₃)

Splittings

Fitting, socle	C₂²×C₃×C₅ ⋊ C₄
derived, Hall	C₃×C₅ ⋊ C₂²×C₄
Sylow	C₅ ⋊ C₂²×Dic₃
Sylow	C₃ ⋊ C₂²×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₄
prime radical	C₂²×C₃×D₅	C₂
cosocle	C₃×D₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×AGL(1,𝔽₅))⋊Dic₃, #91235, C₂×Dic₃×AGL(1,𝔽₅), …
maximal normal subgroup of	(C₂×AGL(1,𝔽₅))⋊Dic₃, #91235, C₂×Dic₃×AGL(1,𝔽₅), …
maximal quotient of	C₂×D₅⋊(C₃⋊C₈), C₂×#6758, D₅⋊#208, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 201);