C₁₉×#19084 #18025043

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

Aliases

direct product

C₁₉×#19084

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

824 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 230 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)(20,21,26,30)(22,31,34,35)(23,33,27,32)(24,29,25,28)(36,37,42,39)(38,46,49,51)(40,45,41,44)(43,50,47,48);(20,22)(21,27)(23,30)(24,32)(25,33)(26,34)(28,35)(29,31)(36,38,42,49)(37,43,39,47)(40,48,41,50)(44,46,45,51);(20,23,26,27)(21,22,30,34)(24,35,25,31)(28,33,29,32)(36,39,42,37)(38,47,49,43)(40,44,41,45)(46,50,51,48);(20,24,26,25)(21,28,30,29)(22,32,34,33)(23,31,27,35)(36,40,42,41)(37,44,39,45)(38,48,49,50)(43,46,47,51)

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, 12072);