C₄₃×#10624798 #44531735

order	31347
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₄₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₄₃×#10624798

Subgroups

	subgroup	G/H
center, socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₄₃
prime radical	C₃×C₄₃×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₄₃ ◁ C₃²×C₉×C₄₃ ◁ G

368 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 61
transitive degree	μ_t(G) = 3483

A minimal faithful permutation representation — 3 generators on 61 points:

(2,4,7)(5,9,8)(10,11,14)(12,16,13)(15,18,17)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(1,2,5,3,4,8,6,7,9);(10,12,16,13,15,18,17,11,14)

Conjugacy classes & character table

3139 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31347, 358);