C₃₁×#1618 #16841551

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#1618

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₃₁×#538	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

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

normal subgroups only

This group has 240 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) = 55
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 3 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)(33,36)(34,38)(37,39)(40,41,45,48)(42,49,51,54)(43,47,44,52)(46,53,55,50);(32,33)(34,39)(35,37)(36,38)(40,42)(41,46)(43,50)(44,53)(45,51)(47,54)(48,55)(49,52);(34,38)(36,39)(40,43,45,44)(41,47,48,52)(42,50,51,53)(46,54,55,49)

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#1618, C₃₁×#18121, C₃₁×#18127, …
maximal normal subgroup of	C₂×C₃₁×#1618, C₃₁×#18121, C₃₁×#18127, …
maximal quotient of	C₃₁×#7790, C₃₁×#7793, C₃₁×#7796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 350);