C₄₃×#10624697 #44531634

order	31347
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₄₃ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₄₃×#10624697

Subgroups

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

368 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 79
transitive degree	μ_t(G) = 10449

A minimal faithful permutation representation — 3 generators on 79 points:

(1,2,4,3,5,7,6,8,9)(10,11,15,13,18,16,14,19,24)(12,17,23,21,27,25,22,28,32)(20,26,31,29,34,33,30,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(10,11,16)(12,17,25)(13,18,24)(14,19,15)(20,26,33)(21,27,32)(22,28,23)(29,34,36)(30,35,31);(10,12,20,14,22,30,13,21,29)(11,17,26,19,28,35,18,27,34)(15,23,31,24,32,36,16,25,33)

Conjugacy classes & character table

6579 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31347, 257);