C₃₁×#2621 #16842554

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#2621

Subgroups

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

528 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 55 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)(36,37,38,39)(41,45)(46,48)(47,53)(54,55);(33,35)(40,41)(42,48)(43,47)(44,45)(46,50)(49,55)(51,54)(52,53);(32,33)(34,35)(36,37)(38,39)(40,42,44,50)(41,46,45,48)(43,51,52,49)(47,55,53,54);(40,43,44,52)(41,47,45,53)(42,49,50,51)(46,54,48,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9195, C₃₁×#9196, C₃₁×#11694, …
maximal normal subgroup of	C₃₁×#9195, C₃₁×#9196, C₃₁×#11694, …
maximal quotient of	C₃₁×#14258, C₃₁×#14261, C₃₁×#14274, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1353);