C₃₁×#2230 #16842163

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₁×#2230

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

126 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 496

A minimal faithful permutation representation — 3 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)(33,36)(34,42,44,39)(35,37)(40,41)(43,45,46,47);(32,33,35,36,38,41,37,40)(34,39,43,45,44,42,46,47);(32,34)(33,39)(35,43)(36,45)(37,46)(38,44)(40,47)(41,42)

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12364, C₃₁×#12365, C₃₁×#12366, …
maximal normal subgroup of	C₃₁×#12364, C₃₁×#12365, C₃₁×#12366, …
maximal quotient of	C₃₁×#12823, C₃₁×#12950, C₃₁×#13116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 962);