C₅₃×#10624559 #50311444

order	38637
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₅₃ · Schur multiplier (H₂) = C₃×C₉

Aliases

direct product

C₅₃×#10624559

Subgroups

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

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 80
transitive degree	μ_t(G) = 38637

A minimal faithful permutation representation — 3 generators on 80 points:

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

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, 119);