C₅₃×#10624780 #50311665

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

Aliases

direct product

C₅₃×#10624780

Subgroups

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

316 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 12879

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

(1,2,4,3,6,8,7,5,9)(10,11,15,13,18,17,14,19,25)(12,20,29,22,28,24,23,16,26)(21,31,35,32,27,36,33,30,34)(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)(10,12,21,14,23,33,13,22,32)(11,16,27,19,28,31,18,20,30)(15,24,34,25,29,36,17,26,35);(10,11,17)(12,16,26)(13,18,25)(14,19,15)(20,29,22)(21,27,35)(23,28,24)(30,36,32)(31,34,33)

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, 340);