C₃₁×#2823 #16842756

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#2823

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

456 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 184 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)(36,37,39,41)(38,40,42,43)(44,45,48,49)(46,47,50,51);(33,35)(38,42)(40,43)(45,49)(47,51);(32,33)(34,35)(36,37)(38,40)(39,41)(42,43)(44,46)(45,47)(48,50)(49,51);(36,38,39,42)(37,40,41,43)(44,47)(45,50)(46,49)(48,51)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12155, C₃₁×#12249, C₂×C₃₁×#2823, …
maximal normal subgroup of	C₃₁×#12155, C₃₁×#12249, C₂×C₃₁×#2823, …
maximal quotient of	C₃₁×#14632, C₃₁×#14633, C₃₁×#14634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1555);