C₃₁×#2388 #16842321

order	3968
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#2388

Subgroups

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

944 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 248 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)(32,33)(34,35)(40,41)(42,45)(43,44)(46,47);(36,37)(38,39)(41,44)(45,47);(33,35)(37,39);(40,42,43,46)(41,45,44,47)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#9009, C₃₁×#9010, C₃₁×#11725, …
maximal normal subgroup of	C₃₁×#9009, C₃₁×#9010, C₃₁×#11725, …
maximal quotient of	C₃₁×#13839, C₃₁×#13844, C₃₁×#13851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1120);