C₃₁×#2816 #16842749

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#2816

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

528 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 1984

A minimal faithful permutation representation — 4 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)(37,40)(41,43)(44,45,48,51)(46,47,50,49);(33,35)(37,40)(38,42)(44,46,48,50)(45,49,51,47);(32,33)(34,35)(36,37)(38,41)(39,40)(42,43)(44,46)(45,49)(47,51)(48,50);(36,38)(37,41)(39,42)(40,43)(44,47,48,49)(45,50,51,46)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9981, C₃₁×#9984, C₃₁×#18451, …
maximal normal subgroup of	C₃₁×#9981, C₃₁×#9984, C₃₁×#18451, …
maximal quotient of	C₃₁×#14622, C₃₁×#14624, C₃₁×#14625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1548);