C₃₁×#2372 #16842305

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃₁×#2372

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

560 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 280 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) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 51 points:

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

Conjugacy classes & character table

1550 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#9439, C₃₁×#9441, C₃₁×#12965, …
maximal normal subgroup of	C₃₁×#9439, C₃₁×#9441, C₃₁×#12965, …
maximal quotient of	C₃₁×#13805, C₃₁×#13816, C₃₁×#13824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1104);