D₄×Q₁₆×C₃₁ #16843219

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×Q₁₆×C₃₁

Subgroups

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

472 conjugacy classes of subgroups.

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)(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)(13,20,19,18,17,16,15,14);(5,13,9,17)(6,14,10,18)(7,15,11,19)(8,16,12,20)

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#12318, C₃₁×#12322, C₃₁×#12370, …
maximal normal subgroup of	C₃₁×#12318, C₃₁×#12322, C₃₁×#12370, …
maximal quotient of	C₃₁×#19811, C₃₁×#19812, C₃₁×#19813, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2018);

D4×Q16×C31 #16843219