C₇₃×C₃⋊ΓL(1,𝔽₁₆) #27190304

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

Aliases

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

Splittings

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

Subgroups

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

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 3285

A minimal faithful permutation representation — 3 generators on 84 points:

(2,3,4,5)(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);(6,7,8)

Conjugacy classes & character table

1533 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 138);

C73×C3⋊ΓL(1,𝔽16) #27190304