C₁₉×#12068 #18018027

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12068

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#442
commutator	C₈	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1580
prime radical	C₁₉×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₁₉×#346 ◁ G

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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)(20,21,25,30,26,31,36,42,27,32,24,35,41,22,33,29)(23,34,37,48,38,28,43,45,39,46,40,44,51,47,49,50);(20,22)(21,26)(23,34)(24,29)(25,42)(27,31)(28,38)(30,33)(32,41)(35,36)(37,44)(39,46)(40,48)(43,50)(45,49)(47,51);(20,23,36,43,41,51,25,37,27,39,33,49,26,38,24,40)(21,28,29,48,22,34,35,50,32,47,42,44,31,46,30,45)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5056);