Q₈×SD₁₆×C₃₁ #16843312

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

Aliases

direct product

Q₈×SD₁₆×C₃₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 47 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#19765, C₃₁×#19769, C₃₁×#19796, …
maximal normal subgroup of	C₃₁×#19765, C₃₁×#19769, C₃₁×#19796, …
maximal quotient of	Q₈×C₃₁×#102, Q₈×C₃₁×#103, C₄⋊C₄×SD₁₆×C₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2111);