Q₈×Q₁₆×C₃₁ #16843315

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

Aliases

direct product

Q₈×Q₁₆×C₃₁

Subgroups

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

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

normal subgroups only

This group has 336 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) = 55
transitive degree	μ_t(G) = 3968

A minimal faithful permutation representation — 4 generators on 55 points:

(1,2,3,4)(5,8,7,6)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12354, C₃₁×#12358, C₃₁×#12406, …
maximal normal subgroup of	C₃₁×#12354, C₃₁×#12358, C₃₁×#12406, …
maximal quotient of	Q₈×C₃₁×#103, C₄⋊C₄×Q₁₆×C₃₁, Q₈×C₃₁×#107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2114);