C₃₁×#1993 #16841926

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#1993

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

276 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 3968

A minimal faithful permutation representation — 3 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)(44,45)(46,50)(47,51)(48,49);(33,35)(37,40)(41,42)(44,46,50,45,48,51,47,49);(36,38,39,43)(37,41,40,42)(45,49)(46,51)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8499, C₃₁×#8999, C₃₁×#9000, …
maximal normal subgroup of	C₃₁×#8499, C₃₁×#8999, C₃₁×#9000, …
maximal quotient of	C₃₁×#9873, C₃₁×#9877, C₃₁×#9888, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 725);