C₄₃×#10624555 #44531492

order	31347
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₄₃ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₄₃×#10624555

Subgroups

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

842 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 70
transitive degree	μ_t(G) = 10449

A minimal faithful permutation representation — 3 generators on 70 points:

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

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, 115);