C₃₁×#2075 #16842008

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#2075

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

256 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 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)(36,37)(38,41)(39,40)(42,43)(44,45,46,49,47,50,51,48);(32,33)(34,35)(37,40)(41,43);(33,35)(36,38,39,42)(37,41,40,43)(45,48)(46,51)(49,50)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7079, C₃₁×#7087, C₃₁×#12278, …
maximal normal subgroup of	C₃₁×#7079, C₃₁×#7087, C₃₁×#12278, …
maximal quotient of	C₃₁×#10615, C₃₁×#10617, C₃₁×#10621, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 807);