C₃₁×#2251 #16842184

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×#2251

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₂×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 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)(33,35)(36,37,41,45)(38,46,48,60)(39,51,53,66)(40,54,52,44)(42,55,56,49)(43,58,59,47)(50,65,64,62)(57,63,61,67);(36,38,47,63,40,50,53,55,41,48,58,67,52,64,39,49)(37,42,51,62,44,57,59,60,45,56,66,65,54,61,43,46);(32,33,34,35)(38,48)(42,56)(46,60)(49,55)(50,64)(57,61)(62,65)(63,67)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12326, C₃₁×#12327, C₃₁×#12338, …
maximal normal subgroup of	C₃₁×#12326, C₃₁×#12327, C₃₁×#12338, …
maximal quotient of	C₃₁×#12507, C₃₁×#12508, C₃₁×#12512, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 983);