C₃₁×#3038 #16842971

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

Aliases

direct product

C₃₁×#3038

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 584 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) = 43
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 4 generators on 43 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)(37,41)(38,39)(42,43);(32,33,34,35);(36,37)(38,41)(39,43)(40,42);(36,38,40,39)(37,41,42,43)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9173, C₃₁×#9174, C₃₁×#9318, …
maximal normal subgroup of	C₃₁×#9173, C₃₁×#9174, C₃₁×#9318, …
maximal quotient of	C₃₁×#19266, C₃₁×#19267, C₃₁×#19268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1770);