C₂₃×#10624451 #30660958

order	16767
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₂₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₂₃×#10624451

Subgroups

	subgroup	G/H
center, socle	C₃³×C₂₃	H(3,𝔽₃)
commutator	C₃³	C₃×C₉×C₂₃
Frattini, cosocle	C₃⁴	C₃²×C₂₃
prime radical	C₃⁴×C₂₃	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₂₃ ◁ C₃⁴×C₂₃ ◁ 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) = 50
transitive degree	μ_t(G) = 5589

A minimal faithful permutation representation — 2 generators on 50 points:

(1,2,4,3,5,7,6,8,9)(11,13,15)(19,20,24,22,25,27,23,26,21)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(10,11,14)(12,15,18)(13,17,16)(19,21,25,22,24,26,23,27,20)

Conjugacy classes & character table

2415 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16767, 11);