C₃₁×#10624724 #36379263

order	22599
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₁ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃₁×#10624724

Subgroups

	subgroup	G/H
center	C₂₇×C₃₁	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₁
Frattini, cosocle	C₃×C₉	C₃³×C₃₁
socle	C₃×C₃₁	C₉×H(3,𝔽₃)
prime radical	C₃₁×#6908	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂₇×C₃₁ ◁ C₃×C₂₇×C₃₁ ◁ G

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

normal subgroups only

This group has 224 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) = 112
transitive degree	μ_t(G) = 2511

A minimal faithful permutation representation — 3 generators on 112 points:

(1,2,6,4,10,19,14,27,39,5,7,18,15,21,38,33,42,60,16,9,25,34,26,48,55,49,68)(3,8,17,12,24,37,30,47,59,13,20,36,31,41,58,53,62,74,32,23,45,54,46,66,71,67,78)(11,22,35,28,44,57,50,65,73,29,40,56,51,61,72,69,75,80,52,43,63,70,64,76,79,77,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112);(2,7,9)(8,20,23)(10,21,26)(22,40,43)(24,41,46)(27,42,49)(44,61,64)(47,62,67)(65,75,77);(1,3,11,4,12,28,14,30,50,5,13,29,15,31,51,33,53,69,16,32,52,34,54,70,55,71,79)(2,8,22,10,24,44,27,47,65,7,20,40,21,41,61,42,62,75,9,23,43,26,46,64,49,67,77)(6,17,35,19,37,57,39,59,73,18,36,56,38,58,72,60,74,80,25,45,63,48,66,76,68,78,81)

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, 284);