C₁₉×#8915 #18014874

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

Aliases

direct product

C₁₉×#8915

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

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

normal subgroups only

This group has 396 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)(20,21,23,22)(24,25,30,35)(26,36,27,38)(28,34,29,33)(31,39,32,37)(41,45)(43,50)(46,54)(47,51)(48,52)(53,55);(20,22,23,21)(24,26,30,27)(25,31,35,32)(28,39,29,37)(33,38,34,36)(40,41,44,47)(42,46,49,55)(43,51,50,45)(48,53,52,54);(24,27,30,26)(25,32,35,31)(28,39,29,37)(33,38,34,36)(40,42,44,49)(41,46,47,55)(43,52,50,48)(45,53,51,54)

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, 1903);