Q₈×#6913 #14517858

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×#6913

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×#6913	C₂²
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈×C₉ ◁ G

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

normal subgroups only

This group has 378 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)(17,22,29)(18,24,30)(23,32,26)(27,31,34)(28,33,35);(1,5,3,7)(2,6,4,8)(10,14,16)(11,17,26)(15,23,22)(18,28,27)(20,29,32)(30,35,34);(9,11,18,12,17,27,19,26,28)(10,15,24,14,22,31,16,23,33)(13,20,30,21,29,34,25,32,35)

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, 2252);