C₁₉×#10624630 #28352826

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

Aliases

direct product

C₁₉×#10624630

Subgroups

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

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

normal subgroups only

This group has 452 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) = 1539

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

(2,4,6)(10,11,14)(12,18,23)(13,17,24)(15,25,21)(16,28,22)(19,30,33)(20,27,34)(26,36,32)(29,35,31)(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)(4,8,7)(10,12,19,13,21,31,22,20,32)(11,15,26,17,27,30,16,18,29)(14,23,33,24,25,35,28,34,36);(12,20,21)(15,18,27)(19,31,32)(23,34,25)(26,30,29)(33,35,36)

Conjugacy classes & character table

1083 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13851, 190);