D₄×C₉⋊H(3,𝔽₃) #14517807

order	1944
nilpotency class	2
isoclinism	isoclinic class of D₄×C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	D₄×C₉⋊H(3,𝔽₃)
unambiguous semidirect product	C₄⋊(C₂×C₉⋊H(3,𝔽₃))
unambiguous semidirect product	(D₄×C₉)⋊H(3,𝔽₃)

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₃×D₄×H(3,𝔽₃)	C₃
LCS	G ▷ C₂×C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

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

normal subgroups only

This group has 664 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) = 22
transitive degree	μ_t(G) = 324

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2201);