C₃₁×#2328 #16842261

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₃₁×#2328

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

740 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 47 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)(33,35)(36,37,39,38)(40,41,44,42)(43,46,47,45);(36,38,39,37)(41,42)(45,46);(32,33,34,35)(40,42,44,41)(43,46,47,45);(40,43,44,47)(41,45,42,46)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#9101, C₃₁×#9105, C₃₁×#9295, …
maximal normal subgroup of	C₃₁×#9101, C₃₁×#9105, C₃₁×#9295, …
maximal quotient of	C₃₁×#13779, C₃₁×#13785, C₃₁×#13791, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1060);