C₃₁×#3152 #16843085

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₁×#3152

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 552 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) = 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)(36,37,40,43)(38,44,45,51)(39,47,48,42)(41,49,50,46);(36,38,39,46,40,45,48,49)(37,41,42,44,43,50,47,51);(32,33,34,35);(38,45)(41,50)(44,51)(46,49)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9421, C₃₁×#9429, C₃₁×#9435, …
maximal normal subgroup of	C₃₁×#9421, C₃₁×#9429, C₃₁×#9435, …
maximal quotient of	C₃₁×#19622, C₃₁×#19626, C₃₁×#19630, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1884);