C₃₁×#1529 #16841462

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×#1529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₃₁×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

216 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 88 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)(33,36,37,38)(40,41,44,47,45,48,43,52)(42,49,50,54,51,46,53,55);(32,33)(34,38)(35,37)(36,39)(40,42)(41,46)(43,50)(44,53)(45,51)(47,54)(48,49)(52,55);(40,43,45,44)(41,47,48,52)(42,50,51,53)(46,54,49,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8412, C₃₁×#8413, C₃₁×#8414, …
maximal normal subgroup of	C₃₁×#8412, C₃₁×#8413, C₃₁×#8414, …
maximal quotient of	C₃₁×#7574, C₃₁×#7577, C₃₁×#7582, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 261);