C₂×ΓL(1,𝔽₃₂)⋊D₃₁ #34063590

order	19220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×ΓL(1,𝔽₃₂)⋊D₃₁
unambiguous semidirect product	(C₂×ΓL(1,𝔽₃₂))⋊D₃₁

Splittings

Fitting, socle	C₂×C₃₁² ⋊ C₂×C₅
derived, Sylow	C₃₁² ⋊ C₂²×C₅
Hall	C₃₁×ΓL(1,𝔽₃₂) ⋊ C₂²
Hall	C₂×C₃₁⋊D₃₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₃₂)⋊D₃₁
commutator, cosocle, hypocenter	C₃₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃₁²	C₂×C₅
LCS	G ▷ C₃₁²
UCS	C₁ ◁ C₂

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 310

A minimal faithful permutation representation — 2 generators on 64 points:

(2,6,7,18,5,3,4,11,12,9)(8,23,24,25,10,14,15,27,29,17)(13,30,22,28,16,20,21,31,19,26)(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);(1,2,5,10,12,19,29,23,7,13,22,30,24,26,14,6,4,8,16,27,21,31,20,11,15,25,28,18,17,9,3)(32,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34)(63,64)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19220, 47);

C2×ΓL(1,𝔽32)⋊D31 #34063590