C₃₁×#2218 #16842151

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#2218

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

170 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 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)(37,41)(38,42)(39,45)(43,47)(44,51)(46,50)(48,49);(36,37,42,49,39,44,46,47,40,43,50,51,45,48,38,41);(33,35)(37,43)(41,47)(44,48)(49,51)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7422, C₃₁×#7424, C₃₁×#13667, …
maximal normal subgroup of	C₃₁×#7422, C₃₁×#7424, C₃₁×#13667, …
maximal quotient of	C₃₁×#12433, C₃₁×#12442, C₃₁×#12454, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 950);