C₁₉×#32805 #18038764

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₉×#32805

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

734 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 2432

A minimal faithful permutation representation — 4 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)(22,25)(23,24)(26,27)(28,29,31,33)(30,32,34,35)(36,37,40,38)(39,44,47,42)(41,43,49,45)(46,50,51,48);(21,24)(22,26)(28,30)(29,32)(31,34)(33,35)(36,38,40,37)(39,46,47,51)(41,43,49,45)(42,48,44,50);(22,26)(25,27)(29,33)(32,35)(36,39,40,47)(37,42,38,44)(41,50,49,48)(43,46,45,51);(20,22,23,26)(21,25,24,27)(30,34)(32,35)(37,38)(42,44)(43,45)(46,51)

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, 25793);