C₁₉×#19096 #18025055

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#19096

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

906 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 218 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)(21,26)(22,29)(23,24)(28,31)(30,32)(33,35)(37,42)(39,40)(43,50)(44,45)(46,48)(49,51);(20,21,25,28)(22,27,30,34)(23,31,24,26)(29,35,32,33)(36,37,41,45)(38,43,47,51)(39,42,40,44)(46,49,48,50);(20,22)(21,27)(23,29)(24,32)(25,30)(26,33)(28,34)(31,35)(36,38)(37,43)(39,46)(40,48)(41,47)(42,49)(44,50)(45,51);(21,28)(26,31)(27,34)(33,35)(36,39,41,40)(37,44,45,42)(38,46,47,48)(43,50,51,49)

Conjugacy classes & character table

589 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 12084);