C₃₁×#3039 #16842972

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#3039

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

480 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 51 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)(33,35)(36,37,41,45)(38,46,48,51)(39,44,40,43)(42,49,50,47);(32,33,34,35);(36,38)(37,42)(39,47)(40,49)(41,48)(43,46)(44,51)(45,50);(36,39,41,40)(37,43,45,44)(38,47,48,49)(42,46,50,51)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9171, C₃₁×#9172, C₃₁×#9316, …
maximal normal subgroup of	C₃₁×#9171, C₃₁×#9172, C₃₁×#9316, …
maximal quotient of	C₃₁×#19270, C₃₁×#19271, C₃₁×#19272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1771);