C₃×C₅₃×#6857 #50311464

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

Aliases

direct product

C₃×C₅₃×#6857

Subgroups

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

680 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 74
transitive degree	μ_t(G) = 12879

A minimal faithful permutation representation — 3 generators on 74 points:

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

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, 139);