C₃₁×#2341 #16842274

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#2341

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

872 conjugacy classes of subgroups; only the 368 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 368 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)(36,37,38,39)(41,42)(45,46);(40,41,44,42)(43,46,47,45);(32,33)(34,35)(36,37)(38,39)(40,42,44,41)(43,46,47,45);(40,43,44,47)(41,45,42,46)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#9364, C₃₁×#9368, C₃₁×#17606, …
maximal normal subgroup of	C₃₁×#9364, C₃₁×#9368, C₃₁×#17606, …
maximal quotient of	C₃₁×#13801, Q₈×C₃₁×#95, C₃₁×#13820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1073);