C₅₃×#10624882 #50311767

order	38637
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₅₃ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₅₃×#10624882

Subgroups

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

1058 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 476 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 — 4 generators on 89 points:

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

Conjugacy classes & character table

5565 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 442);