C₃₁×#2503 #16842436

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#2503

Subgroups

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

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 1984

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

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#9460, C₃₁×#9953, C₃₁×#9961, …
maximal normal subgroup of	C₃₁×#9460, C₃₁×#9953, C₃₁×#9961, …
maximal quotient of	C₃₁×#13881, C₃₁×#13904, C₃₁×#13972, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1235);