C₃₁×#2812 #16842745

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃₁×#2812

Subgroups

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

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

normal subgroups only

This group has 712 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) = 992

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)(46,49)(47,48)(50,51);(33,35)(37,40)(38,42)(45,48)(46,50);(32,33)(34,35)(36,37)(38,41)(39,40)(42,43)(44,45)(46,51)(47,48)(49,50);(36,38)(37,41)(39,42)(40,43)(44,46)(45,49)(47,50)(48,51)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12093, C₃₁×#12205, C₂×C₃₁×#2812, …
maximal normal subgroup of	C₃₁×#12093, C₃₁×#12205, C₂×C₃₁×#2812, …
maximal quotient of	C₃₁×#14621, C₃₁×#14622, C₃₁×#14627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1544);