C₃₁×#3058 #16842991

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₁×#3058

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×#41×C₃₁ ◁ G

524 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 992

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)(32,33)(34,35)(37,41)(39,42)(43,47)(44,48)(46,49)(50,51);(36,37,42,48,40,44,39,41)(38,43,49,51,45,50,46,47);(33,35)(36,38)(37,43)(39,46)(40,45)(41,47)(42,49)(44,50)(48,51);(38,45)(43,50)(46,49)(47,51)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#11868, C₃₁×#11870, C₃₁×#13074, …
maximal normal subgroup of	C₃₁×#11868, C₃₁×#11870, C₃₁×#13074, …
maximal quotient of	C₃₁×#19187, C₃₁×#19192, C₃₁×#19198, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1790);