C₃₁×#2240 #16842173

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#2240

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₃₁×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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)(34,37)(35,36)(38,39)(41,47)(42,52)(43,54)(44,55)(45,48)(49,50)(51,53);(33,36)(37,39)(40,41,44,49,45,50,43,47,46,51,54,42,48,52,55,53);(32,34,35,38)(33,37,36,39)(40,42,43,41,48,47,44,52,46,49,55,51,45,53,54,50)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12340, C₃₁×#12341, C₃₁×#12342, …
maximal normal subgroup of	C₃₁×#12340, C₃₁×#12341, C₃₁×#12342, …
maximal quotient of	C₃₁×#12888, C₃₁×#13147, C₃₁×#13202, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 972);