C₁₉×#10319 #18016278

order	4864
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#10319

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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)(22,23)(25,30)(27,28)(31,34)(32,33)(35,36)(38,39)(40,41,45,49)(42,50,53,54)(43,48,44,47)(46,52,55,51);(21,23)(24,25,29,33)(26,34,37,38)(27,32,28,30)(31,35,39,36)(40,42,45,53)(41,46,49,55)(43,52,44,51)(47,54,48,50);(24,26,29,37)(25,31,33,39)(27,36,28,35)(30,34,32,38)(41,47)(43,44)(46,54)(48,49)(50,55)(51,52)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 3307);