C₃₁×#2162 #16842095

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#2162

Subgroups

	subgroup	G/H
center	C₄×C₈×C₃₁	C₂²
commutator	C₂	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂²×C₈
prime radical	C₂³×C₃₁	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₃₁ ◁ G

160 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 120 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)(40,41,42,45,43,46,48,51,44,47,49,52,50,53,54,55);(33,36)(37,39);(32,34,35,38)(33,37,36,39)

Conjugacy classes & character table

2480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7142, C₃₁×#7144, C₃₁×#7145, …
maximal normal subgroup of	C₃₁×#7142, C₃₁×#7144, C₃₁×#7145, …
maximal quotient of	C₃₁×#12590, C₄×C₃₁×C₂²⋊C₁₆, C₄×C₃₁×C₄⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 894);