C₅₃×#10624755 #50311640

order	38637
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₅₃ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₅₃×#10624755

Subgroups

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

446 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1431

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

(1,2,5)(3,9,19)(4,7,15)(6,14,11)(8,18,13)(10,20,23)(12,17,26)(16,24,22)(21,25,27)(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);(1,3,10,4,12,21,13,11,22)(2,6,16,7,9,20,8,17,25)(5,14,23,15,19,27,18,26,24);(2,7,8)(6,9,17)(16,20,25)

Conjugacy classes & character table

3021 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 315);