C₁₉×#32868 #18038827

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#32868

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

596 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 206 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)(20,21,23,26)(22,25,27,24)(28,29)(30,33)(31,32)(34,35)(36,37)(38,41)(39,42)(40,43)(44,51)(45,49)(46,48)(47,50);(20,22)(21,24)(23,27)(25,26)(29,32)(33,35)(36,38,39,45)(37,41,42,49)(40,44,47,46)(43,51,50,48);(20,22)(21,25)(23,27)(24,26)(29,32)(30,34)(37,42)(38,44)(40,47)(41,48)(45,46)(49,51);(22,27)(24,25)(28,30,31,34)(29,33,32,35)(37,42)(38,45)(43,50)(44,46)

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, 25856);