C₁₉×#20319 #18026278

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

Aliases

direct product

C₁₉×#20319

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#493	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ G

578 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 198 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)(21,25)(23,31)(26,32)(27,34)(28,35)(29,33)(36,37)(38,44)(39,47)(40,42)(41,46)(43,49)(45,51)(48,50);(20,21,24,26)(22,28,30,34)(23,32,31,25)(27,33,35,29)(38,45)(41,50)(44,51)(46,48);(21,26)(25,32)(27,35)(28,34)(36,38,40,45)(37,41,42,50)(39,48,49,46)(43,51,47,44);(20,22,24,30)(21,27,26,35)(23,33,31,29)(25,34,32,28)(37,42)(41,50)(43,47)(44,51)

Conjugacy classes & character table

703 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 13307);