C₃₁×#1517 #16841450

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#1517

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 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)(32,33)(34,38)(35,37)(36,39)(40,41)(42,48,50,45)(43,46)(44,47)(49,54,51,53)(52,55);(33,37)(34,35)(38,39)(40,42)(41,45)(43,51)(44,50)(46,54)(47,48)(49,52)(53,55);(32,34,36,35)(33,37,39,38)(40,43,44,52)(41,46,47,55)(42,49,50,51)(45,53,48,54)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8428, C₃₁×#8429, C₃₁×#8430, …
maximal normal subgroup of	C₃₁×#8428, C₃₁×#8429, C₃₁×#8430, …
maximal quotient of	C₃₁×#7574, C₃₁×#7587, C₃₁×#7588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 249);