D₄×#6887 #14517802

order	1944
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₉ · Schur multiplier (H₂) = C₂×C₃×C₉

Aliases

direct product	D₄×#6887
unambiguous semidirect product	C₄⋊(C₂×#6887)

Subgroups

	subgroup	G/H
center	C₂×C₃×C₉	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃²×C₉
Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₃³×D₄	C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ G

552 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 324 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)(5,6,8,7,9,11,10,12,13)(14,15,18)(16,20,26)(17,21,25)(19,27,24)(22,29,34)(23,30,33)(28,35,32)(31,37,39)(36,40,38);(2,4)(15,19,21)(18,25,27)(20,28,30)(26,33,35)(29,36,37)(34,39,40);(14,16,22,17,23,31,24,32,38)(15,20,29,21,30,37,19,28,36)(18,26,34,25,33,39,27,35,40)

Conjugacy classes & character table

495 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2196);