Q₈×#6908 #14517853

order	1944
nilpotency class	3
isoclinism	isoclinic class of D₄×AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product

Q₈×#6908

Subgroups

	subgroup	G/H
center	C₂×C₉	C₂²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
prime radical	C₂×#6908	C₂²
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₃×Q₈×C₉ ◁ G

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

normal subgroups only

This group has 402 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)(11,15,20)(12,14,21)(16,25,19)(17,23,28)(18,22,29)(24,32,27)(26,30,34)(31,35,33);(1,5,3,7)(2,6,4,8)(10,14,16)(15,22,24)(23,30,31);(9,11,17,12,18,26,19,27,33)(10,15,23,14,22,30,16,24,31)(13,20,28,21,29,34,25,32,35)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2247);