C₃₁×#3080 #16843013

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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)(32,33,35,37)(34,38,39,36)(40,41,44,46)(42,50,52,55)(43,49,45,48)(47,53,54,51);(32,34,35,39)(33,36,37,38)(41,46)(47,54)(48,49)(50,55);(40,42)(41,47)(43,51)(44,52)(45,53)(46,54)(48,50)(49,55);(40,43,44,45)(41,48,46,49)(42,51,52,53)(47,50,54,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9351, C₃₁×#9352, C₃₁×#24724, …
maximal normal subgroup of	C₃₁×#9351, C₃₁×#9352, C₃₁×#24724, …
maximal quotient of	C₃₁×#20649, C₃₁×#20875, C₃₁×#21034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1812);