C₁₉×#32710 #18038669

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

Aliases

direct product

C₁₉×#32710

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁵×C₁₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂⁴×C₄×C₁₉ ◁ G

984 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 1216

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

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 25698);