C₃₁×#2216 #16842149

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2216

Subgroups

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

154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 67
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 67 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)(36,37,41,45)(38,46,50,62)(39,51,53,66)(40,54,52,44)(42,55,57,48)(43,58,59,47)(49,65,64,61)(56,63,60,67);(36,38,47,63,40,49,53,55,41,50,58,67,52,64,39,48)(37,42,51,61,44,56,59,62,45,57,66,65,54,60,43,46);(33,35)

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7419, C₃₁×#7421, C₃₁×#13664, …
maximal normal subgroup of	C₃₁×#7419, C₃₁×#7421, C₃₁×#13664, …
maximal quotient of	C₃₁×#12431, C₃₁×#12440, C₃₁×#12452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 948);