C₃₁×#1735 #16841668

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#1735

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 55 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)(36,37,38,39)(40,41)(42,50)(43,47)(44,48)(45,49)(46,51)(52,55)(53,54);(33,35)(40,42,44,52,45,53,43,51)(41,46,47,54,49,55,48,50);(40,43,45,44)(41,47,49,48)(42,51,53,52)(46,54,55,50)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7260, C₃₁×#7262, C₃₁×#10706, …
maximal normal subgroup of	C₃₁×#7260, C₃₁×#7262, C₃₁×#10706, …
maximal quotient of	C₃₁×#7872, C₃₁×#7875, C₃₁×#7878, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 467);