D₄×#6896 #14517811

order	1944
nilpotency class	2
isoclinism	isoclinic class of D₄×C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₉

Aliases

direct product	D₄×#6896
unambiguous semidirect product	C₄⋊(C₂×#6896)

Subgroups

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

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

normal subgroups only

This group has 480 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.

Full subgroup data →

Presentation

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

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, 2205);