C₁₉×#9004 #18014963

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#9004

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#120
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 55
transitive degree	μ_t(G) = 4864

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,28,32,30,33,29,38)(26,34,35,39,27,36,37,31)(40,41)(42,48,50,55)(43,45)(44,47)(46,49,53,52)(51,54);(20,21,22,23)(24,26,30,27)(25,31,33,39)(28,37,29,35)(32,36,38,34)(41,45)(44,51)(46,53)(49,52);(24,27,30,26)(25,31,33,39)(28,37,29,35)(32,34,38,36)(40,42)(41,46)(43,50)(44,52)(45,53)(47,55)(48,54)(49,51)

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, 1992);