Q₈×C₉⋊H(3,𝔽₃) #14517837

order	1944
nilpotency class	2
isoclinism	isoclinic class of D₄×C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

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

498 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2,3,4)(5,8,7,6)(10,12,15)(13,17,16)(18,19,21)(20,23,25)(22,26,24);(1,5,3,7)(2,6,4,8)(9,10,13,11,12,16,14,15,17);(19,22,23)(21,25,26)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2231);