C₁₉×#10624827 #28353023

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

Aliases

direct product

C₁₉×#10624827

Subgroups

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

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

normal subgroups only

This group has 216 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) = 73
transitive degree	μ_t(G) = 4617

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,5)(3,7,13)(4,8,12)(6,14,11)(9,16,21)(10,17,20)(15,22,19)(18,24,26)(23,27,25)(28,29,33)(30,36,46)(31,39,32)(34,42,50)(35,45,40)(37,48,52)(38,44,41)(43,53,49)(47,51,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27)(28,30,37,32,38,49,40,50,54)(29,34,43,31,36,47,35,44,48)(33,41,51,39,42,52,45,46,53);(1,3,9,4,10,18,11,19,25)(2,7,16,8,17,24,6,15,23)(5,13,21,12,20,26,14,22,27)(28,31,39,32,35,45,40,29,33)(30,36,42,38,44,46,50,34,41)(37,47,52,49,48,53,54,43,51)

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, 387);