H(3,𝔽₃)×3₋¹⁺²×C₄₃ #44531832

order	31347
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₄₃ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

H(3,𝔽₃)×3₋¹⁺²×C₄₃

Subgroups

	subgroup	G/H
center, socle	C₃²×C₄₃	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₄₃
prime radical	C₃²×H(3,𝔽₃)×C₄₃	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₄₃ ◁ G

1244 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 518 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 — 4 generators on 61 points:

(1,2,4)(3,6,8)(5,9,7)(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);(2,5,6)(4,8,9);(10,11,13,12,15,17,16,14,18);(11,14,15)(13,17,18)

Conjugacy classes & character table

5203 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31347, 455);