C₁₉×#10624613 #28352809

order	13851
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₁₉ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₁₉×#10624613

Subgroups

	subgroup	G/H
center, socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₉
prime radical	C₁₉×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃²×C₉×C₁₉ ◁ G

314 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 4617

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

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

Conjugacy classes & character table

1387 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13851, 173);