C₄₁×#10624449 #44470884

order	29889
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₄₁ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₄₁×#10624449

Subgroups

	subgroup	G/H
center, socle	C₃³×C₄₁	H(3,𝔽₃)
commutator	C₃³	C₃×C₉×C₄₁
Frattini, cosocle	C₃⁴	C₃²×C₄₁
prime radical	C₃²×H(3,𝔽₃)×C₄₁	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₄₁ ◁ C₃⁴×C₄₁ ◁ G

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

normal subgroups only

This group has 458 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) = 68
transitive degree	μ_t(G) = 9963

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

4305 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29889, 9);