C₃₁×#2312 #16842245

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

Aliases

direct product

C₃₁×#2312

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

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

normal subgroups only

This group has 672 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) = 992

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)(32,33,34,35)(36,37)(38,42)(39,40)(41,43)(45,48)(46,50);(37,40)(41,42)(44,45,47,48)(46,49,50,51);(36,37)(38,41)(39,40)(42,43)(44,45)(46,49)(47,48)(50,51);(36,38)(37,41)(39,43)(40,42)(44,46)(45,49)(47,50)(48,51)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8736, C₃₁×#8750, C₃₁×#8759, …
maximal normal subgroup of	C₃₁×#8736, C₃₁×#8750, C₃₁×#8759, …
maximal quotient of	C₃₁×#13775, C₃₁×#13778, C₃₁×#13788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1044);