C₃₁×#1518 #16841451

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1518

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	C₂³×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 220 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)(33,37)(34,35)(38,39)(40,41)(42,48,50,45)(43,46)(44,47)(49,54,51,53)(52,55);(32,33,36,39)(34,37,35,38)(40,42,44,50)(41,45,47,48)(43,51,52,49)(46,54,55,53);(32,34,36,35)(33,38,39,37)(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×#8430, C₃₁×#8431, C₃₁×#8432, …
maximal normal subgroup of	C₃₁×#8430, C₃₁×#8431, C₃₁×#8432, …
maximal quotient of	C₃₁×#7577, C₃₁×#7590, C₃₁×#7591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 250);