Q₈×#6910 #14517855

order	1944
nilpotency class	3
isoclinism	isoclinic class of D₄×#6909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product

Q₈×#6910

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₂²×C₃
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃³×Q₈ ◁ G

408 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3,4)(5,8,7,6)(9,10,13)(11,17,22)(12,16,23)(14,24,20)(15,27,21)(18,29,32)(19,26,33)(25,35,31)(28,34,30);(1,5,3,7)(2,6,4,8)(9,11,18,12,20,30,21,19,31)(10,14,25,16,26,29,15,17,28)(13,22,32,23,24,34,27,33,35);(11,19,20)(14,17,26)(18,30,31)(22,33,24)(25,29,28)(32,34,35)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2249);