C₂×(C₄₁⋊C₅)⋊D₄₁ #46777436

order	33620
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×(C₄₁⋊C₅)⋊D₄₁
unambiguous semidirect product	(C₂×C₄₁⋊C₅)⋊D₄₁

Splittings

Fitting, socle	C₂×C₄₁² ⋊ C₂×C₅
derived, Sylow	C₄₁² ⋊ C₂²×C₅
Hall	C₄₁×C₄₁⋊C₅ ⋊ C₂²
Hall	C₂×C₄₁⋊D₄₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₄₁⋊C₅)⋊D₄₁
commutator, cosocle, hypocenter	C₄₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₄₁²	C₂×C₅
LCS	G ▷ C₄₁²
UCS	C₁ ◁ C₂

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 410

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33620, 71);

C2×(C41⋊C5)⋊D41 #46777436