C₃₁×#3025 #16842958

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#3025

Subgroups

	subgroup	G/H
center	C₄×C₃₁	#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#120
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

766 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 212 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) = 496

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

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#13045, C₃₁×#13046, C₃₁×#21483, …
maximal normal subgroup of	C₃₁×#13045, C₃₁×#13046, C₃₁×#21483, …
maximal quotient of	C₃₁×#18122, C₃₁×#18123, C₃₁×#18124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1757);