C₅₃×#10624879 #50311764

order	38637
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₅₃ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₅₃×#10624879

Subgroups

	subgroup	G/H
center	C₃×C₉×C₅₃	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₅₃
socle	C₃²×C₅₃	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₅₃ ◁ G

758 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 494 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 — 4 generators on 89 points:

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

Conjugacy classes & character table

8109 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 439);