C₃₁×#2003 #16841936

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#2003

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂×C₃₁×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

382 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 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)(36,37)(38,42)(39,41)(40,44)(43,47)(45,50)(46,51)(48,49);(37,41)(39,47)(42,50)(43,44)(45,48)(49,51);(33,35)(36,38,40,46)(37,42,44,51)(39,48,47,45)(41,49,43,50)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2003, C₃₁×#23399, C₃₁×#23400, …
maximal normal subgroup of	C₂×C₃₁×#2003, C₃₁×#23399, C₃₁×#23400, …
maximal quotient of	C₃₁×#9915, C₃₁×#9921, C₃₁×#9923, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 735);