C₂×ΓL(1,𝔽₃₂)⋊D₆₁ #49180682

order	37820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×ΓL(1,𝔽₃₂)⋊D₆₁
unambiguous semidirect product	(C₂×ΓL(1,𝔽₃₂))⋊D₆₁

Splittings

Fitting, socle	C₂×C₃₁×C₆₁ ⋊ C₂×C₅
derived, Hall	C₃₁×C₆₁ ⋊ C₂²×C₅
Hall	C₆₁×ΓL(1,𝔽₃₂) ⋊ C₂²
Hall	C₂×D₁₈₉₁ ⋊ C₅
Hall	ΓL(1,𝔽₃₂) ⋊ C₂×D₆₁
Sylow	C₆₁ ⋊ C₂×D₃₁⋊C₅
Sylow	C₃₁ ⋊ C₂×C₅×D₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₃₂)⋊D₆₁
commutator, cosocle, hypocenter	C₃₁×C₆₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃₁×C₆₁	C₂×C₅
LCS	G ▷ C₃₁×C₆₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 3782

A minimal faithful permutation representation — 2 generators on 94 points:

(4,8,9,20,7,5,6,13,14,11)(10,25,26,27,12,16,17,29,31,19)(15,32,24,30,18,22,23,33,21,28)(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)(85,86)(87,88)(89,90)(91,92)(93,94);(1,2)(3,4,7,12,14,21,31,25,9,15,24,32,26,28,16,8,6,10,18,29,23,33,22,13,17,27,30,20,19,11,5)(34,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37820, 62);

C2×ΓL(1,𝔽32)⋊D61 #49180682