C₃₁×#10624728 #36379267

order	22599
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₁ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃₁×#10624728

Subgroups

	subgroup	G/H
center	C₃×C₉×C₃₁	C₃³
commutator	C₃²	C₃²×C₉×C₃₁
Frattini, cosocle	C₃×C₉	C₃³×C₃₁
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃³×C₃₁	C₃×C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₁ ◁ G

192 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 67
transitive degree	μ_t(G) = 7533

A minimal faithful permutation representation — 3 generators on 67 points:

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

Conjugacy classes & character table

4743 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22599, 288);