C₃×ΓL(1,𝔽₁₆)×C₇₃ #27190292

order	13140
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×ΓL(1,𝔽₁₆)×C₇₃
unambiguous semidirect product	C₃⋊(C₃×AGL(1,𝔽₅)×C₇₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₇₃ ⋊ C₄
derived	C₃×C₅ ⋊ C₃×C₄×C₇₃
Hall	C₃²×C₇₃ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₇₃ ⋊ C₃×Dic₃
Hall	C₃²×C₅ ⋊ C₄×C₇₃
Sylow	C₃² ⋊ AGL(1,𝔽₅)×C₇₃
Sylow	C₅ ⋊ C₃×Dic₃×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₃×C₄×C₇₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇₃	C₄
prime radical	C₃²×D₅×C₇₃	C₂
cosocle	C₃×D₅	C₂×C₃×C₇₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₇₃

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 2190

A minimal faithful permutation representation — 2 generators on 84 points:

(2,5,4,3)(6,7,8)(10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(1,2,5,3,4)(9,10,11)

Conjugacy classes & character table

1971 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 126);

C3×ΓL(1,𝔽16)×C73 #27190292