C₃₁×#1531 #16841464

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

Aliases

direct product

C₃₁×#1531

Subgroups

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

212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 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)(33,37,38,39)(40,41,43,47,44,46,52,54)(42,48,49,55,50,45,51,53);(32,33)(34,37)(35,39)(36,38)(40,42,44,50)(41,45,46,48)(43,51,52,49)(47,55,54,53);(32,34,36,35)(33,37,38,39)(41,46)(45,48)(47,54)(53,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8417, C₃₁×#8418, C₃₁×#8457, …
maximal normal subgroup of	C₃₁×#8417, C₃₁×#8418, C₃₁×#8457, …
maximal quotient of	C₃₁×#7579, C₃₁×#7588, C₃₁×#7591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 263);