D₄×C₁₆×C₃₁ #16842100

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×C₁₆×C₃₁

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂²
commutator	C₂	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂²×C₈
prime radical	C₂×D₄×C₃₁	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

174 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 51 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(2,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×D₄⋊C₃₂, C₁₆×D₈×C₃₁, C₃₁×#12038, …
maximal normal subgroup of	C₃₁×D₄⋊C₃₂, C₁₆×D₈×C₃₁, C₃₁×#12038, …
maximal quotient of	C₁₆×D₈×C₃₁, C₁₆×SD₁₆×C₃₁, C₁₆×Q₁₆×C₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 899);