C₃₁×#1749 #16841682

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#1749

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂³
commutator	C₂²	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

412 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 3 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)(32,33)(34,35)(36,37,39,38);(33,35)(40,41,42,45,43,44,46,47);(36,38,39,37)(41,44)(45,47)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7021, C₃₁×#7033, C₃₁×#7036, …
maximal normal subgroup of	C₃₁×#7021, C₃₁×#7033, C₃₁×#7036, …
maximal quotient of	C₃₁×#8145, C₃₁×#8147, C₃₁×#8152, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 481);