C₁₉×#10624612 #28352808

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

Aliases

direct product

C₁₉×#10624612

Subgroups

	subgroup	G/H
center, socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₉
prime radical	C₃⁴×C₁₉	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃⁴×C₁₉ ◁ G

356 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 1539

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

(2,4,6)(10,11,14,21,16,27,13,17,23)(12,18,22,31,26,34,20,15,24)(19,29,32,36,25,35,30,28,33)(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)(10,12,19,21,31,36,13,20,30)(11,15,25,16,18,28,17,26,29)(14,22,32,27,34,35,23,24,33);(11,16,17)(14,23,27)(15,18,26)(22,24,34)(25,28,29)(32,33,35)

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, 172);