C₅₃×#10624556 #50311441

order	38637
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₅₃ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₅₃×#10624556

Subgroups

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

554 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 12879

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

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

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, 116);