D₄×#6899 #14517814

order	1944
nilpotency class	2
isoclinism	isoclinic class of D₄×C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₄×#6899
unambiguous semidirect product	C₄⋊(C₂×#6899)

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

440 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 40
transitive degree	μ_t(G) = 324

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4)(6,9,7)(10,11,12)(14,15,19)(16,23,35)(17,21,29)(18,22,30)(20,31,26)(24,34,38)(25,36,28)(27,33,39)(32,40,37);(2,4)(5,6,10,13,7,11,8,9,12)(14,16,24,18,25,37,27,26,17)(15,20,32,22,23,21,33,36,34)(19,28,29,30,31,38,39,35,40);(5,7,12,13,9,10,8,6,11)(14,17,26,27,37,25,18,24,16)(15,21,20,33,32,36,22,34,23)(19,29,31,39,40,28,30,38,35)

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, 2208);