C₁₉×#7503 #18013462

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7503

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#354
commutator	C₄²	C₄²×C₁₉
Frattini, cosocle	#532	C₂²×C₁₉
socle	C₂×C₁₉	#1393
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,28)(22,31,33,48)(24,29,38,46)(25,30,36,47)(26,40,41,50)(27,37,44,39)(32,43,49,45)(34,42,35,51);(20,22)(21,26)(23,35,36,40)(24,34)(25,32)(27,43)(28,45,47,48)(29,31)(30,42)(33,38,49,37)(39,50)(41,44,51,46)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 491);