C₄₁×#239 #14520245

order	1968
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄₁ · Schur multiplier (H₂) = C₂²

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₄₁×#239

Splittings

Fitting	C₃×Q₈×C₄₁ ⋊ C₂
Hall	C₃×C₄₁ ⋊ #41
Sylow	C₃ ⋊ #41×C₄₁

Subgroups

	subgroup	G/H
center	C₂×C₄₁	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₄₁
Frattini	C₂	C₂²×D₃×C₄₁
Fitting	C₃×Q₈×C₄₁	C₂
socle	C₂×C₃×C₄₁	C₂³
hypercenter	Q₈×C₄₁	D₃
hypocenter	C₃	#41×C₄₁
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄₁ ◁ Q₈×C₄₁

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 984

A minimal faithful permutation representation — 3 generators on 52 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1968, 168);

C41×#239 #14520245