C₃₁×#2616 #16842549

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#2616

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

816 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 47 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)(37,39)(40,41)(42,44)(43,45)(46,47);(33,35)(37,39)(41,42)(45,46);(32,33)(34,35)(36,37)(38,39)(40,42,44,41)(43,45,47,46);(40,43,44,47)(41,45,42,46)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9931, C₃₁×#9935, C₃₁×#18230, …
maximal normal subgroup of	C₃₁×#9931, C₃₁×#9935, C₃₁×#18230, …
maximal quotient of	C₃₁×#14258, C₃₁×#14264, C₃₁×#14300, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1348);