C₅₃×#10624783 #50311668

order	38637
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₅₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₅₃×#10624783

Subgroups

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

368 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 89
transitive degree	μ_t(G) = 4293

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,4,3,6,8,7,5,9)(10,11,14)(12,18,28)(13,17,23)(15,25,22)(16,24,21)(19,29,32)(20,26,33)(27,35,31)(30,34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,5,6)(4,8,9)(11,15,17)(14,23,25)(16,26,18)(24,28,33)(27,34,29)(32,36,35);(10,12,19,13,20,30,22,21,31)(11,16,27,17,18,29,15,26,34)(14,24,32,23,28,36,25,33,35)

Conjugacy classes & character table

3869 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 343);