C₃₁×#2220 #16842153

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2220

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

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

normal subgroups only

This group has 160 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)(38,45)(39,46)(42,47)(43,48)(44,51)(49,50);(36,37,42,51,39,44,38,41,40,43,45,50,46,49,47,48);(33,35)(37,43)(41,48)(44,49)(50,51)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2220, C₃₁×#33604, C₃₁×#33605, …
maximal normal subgroup of	C₂×C₃₁×#2220, C₃₁×#33604, C₃₁×#33605, …
maximal quotient of	C₃₁×#12435, C₃₁×#12444, C₃₁×#12456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 952);