C₁₉×#20321 #18026280

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

Aliases

direct product

C₁₉×#20321

Subgroups

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

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

normal subgroups only

This group has 542 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) = 4864

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)(20,21,26,24)(22,29,32,35)(23,28,25,33)(27,30,34,31)(36,37)(38,44)(39,47)(40,42)(41,46)(43,49)(45,51)(48,50);(20,22,26,32)(21,27,24,34)(23,31,25,30)(28,35,33,29)(38,45)(41,50)(44,51)(46,48);(20,23,26,25)(21,28,24,33)(22,30,32,31)(27,29,34,35)(36,38,40,45)(37,41,42,50)(39,48,49,46)(43,51,47,44);(20,24,26,21)(22,29,32,35)(23,33,25,28)(27,30,34,31)(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, 13309);