C₃₁×#2316 #16842249

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

Aliases

direct product

C₃₁×#2316

Subgroups

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

600 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 360 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)(40,41)(42,46)(43,45)(44,47);(40,41,43,45)(42,44,47,46);(32,33,34,35)(40,41)(42,44)(43,45)(46,47);(40,42)(41,44)(43,47)(45,46)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8757, C₃₁×#8856, C₃₁×#8857, …
maximal normal subgroup of	C₃₁×#8757, C₃₁×#8856, C₃₁×#8857, …
maximal quotient of	C₃₁×#13778, C₃₁×#13783, C₃₁×#13795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1048);