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C₅×ΓL(1,𝔽₁₆) #64489

order	300
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = C₄×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×ΓL(1,𝔽₁₆)
unambiguous semidirect product	C₃⋊(C₅×AGL(1,𝔽₅))

Splittings

Fitting, socle, Hall	C₃×C₅² ⋊ C₄
derived	C₃×C₅ ⋊ C₄×C₅
Sylow	C₅² ⋊ Dic₃
Sylow	C₃ ⋊ C₅×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, hypercenter	C₅	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₄
prime radical	C₃×C₅×D₅	C₂
cosocle	C₃×D₅	C₂×C₅
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×AGL(1,𝔽₅), D₅×ΓL(1,𝔽₁₆), ΓL(1,𝔽₁₆)⋊D₅, …
maximal normal subgroup of	C₅×D₃×AGL(1,𝔽₅), D₅×ΓL(1,𝔽₁₆), ΓL(1,𝔽₁₆)⋊D₅, …
maximal quotient of	C₅×C₃⋊#170, C₂×C₅×ΓL(1,𝔽₁₆), C₅×C₉⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 32);