C₁₉×#8759 #18014718

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#8759

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#381
commutator	C₂⁴	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

474 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 51 points:

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

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1747);