C₃₁×#2683 #16842616

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#2683

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

592 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 224 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) = 47
transitive degree	μ_t(G) = 1984

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

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7738, C₃₁×#7754, C₃₁×#7764, …
maximal normal subgroup of	C₃₁×#7738, C₃₁×#7754, C₃₁×#7764, …
maximal quotient of	C₃₁×#14362, C₃₁×#14401, C₃₁×#14440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1415);