C₃₁×#388 #14520569

order	1984
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#388

Subgroups

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

190 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 43
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 3 generators on 43 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)(33,35)(37,38)(40,41,43,42);(36,37,39,38)(40,42,43,41);(32,33)(34,35)(36,38,39,37)

Conjugacy classes & character table

868 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃₁×#388, #16848153, #16848171, …
maximal subgroup of	C₂×C₃₁×#388, C₃₁×#2290, C₃₁×#2293, …
maximal normal subgroup of	C₂×C₃₁×#388, C₃₁×#2290, C₃₁×#2293, …
maximal quotient of	C₃₁×#1434, C₃₁×#1436, C₃₁×#1437, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1984, 69);