C₁₉×#19006 #18024965

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

Aliases

direct product

C₁₉×#19006

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#562	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

938 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 222 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)(22,30)(23,29)(24,28)(25,27)(26,31)(32,35)(33,34)(36,37,40,42)(38,44,46,51)(39,47,49,43)(41,48,50,45);(20,22)(21,26)(23,33)(24,31)(25,32)(27,34)(28,35)(29,30)(36,38,40,46)(37,41,42,50)(39,48,49,45)(43,51,47,44);(20,22)(21,26)(23,31)(24,33)(25,32)(27,34)(28,30)(29,35)(39,49)(43,47)(44,51)(45,48);(21,27)(26,34)(28,29)(30,35)(37,42)(41,50)(43,47)(44,51)

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, 11994);