C₅×C₄₃×ΓL(1,𝔽₁₆) #27166453

order	12900
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₄₃×ΓL(1,𝔽₁₆)
unambiguous semidirect product	C₃⋊(C₅×AGL(1,𝔽₅)×C₄₃)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₄₃ ⋊ C₄
derived	C₃×C₅ ⋊ C₄×C₅×C₄₃
Hall	C₅²×C₄₃ ⋊ Dic₃
Hall	C₃×C₄₃ ⋊ C₅×AGL(1,𝔽₅)
Hall	C₃×C₅² ⋊ C₄×C₄₃
Sylow	C₅² ⋊ Dic₃×C₄₃
Sylow	C₃ ⋊ C₅×AGL(1,𝔽₅)×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₄₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₅×C₄₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₄₃	C₄
prime radical	C₃×C₅×D₅×C₄₃	C₂
cosocle	C₃×D₅	C₂×C₅×C₄₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅×C₄₃

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 2580

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2,3,4,5)(7,10,9,8)(12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56);(6,7,10,8,9)(11,12,13)

Conjugacy classes & character table

1935 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 122);

C5×C43×ΓL(1,𝔽16) #27166453