C₃₁×#10624450 #36378989

order	22599
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₃₁ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₃₁×#10624450

Subgroups

	subgroup	G/H
center, socle	C₃³×C₃₁	H(3,𝔽₃)
commutator	C₃³	C₃×C₉×C₃₁
Frattini, cosocle	C₃⁴	C₃²×C₃₁
prime radical	C₃⁴×C₃₁	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₃₁ ◁ C₃⁴×C₃₁ ◁ G

254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 58
transitive degree	μ_t(G) = 7533

A minimal faithful permutation representation — 2 generators on 58 points:

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

Conjugacy classes & character table

3255 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22599, 10);