D₄×#6883 #14516249

order	1944
nilpotency class	4
isoclinism	isoclinic class of D₄×#6881
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₄×#6883
unambiguous semidirect product	C₄⋊(C₂×#6883)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#732
commutator, Frattini, cosocle	C₂×C₃×C₉	C₂²×C₃²
prime radical	D₄×#732	C₃
LCS	G ▷ C₂×C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×D₄ ◁ C₃×D₄×C₉ ◁ G

160 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3,4)(5,6,9,17,12,25,8,13,19)(7,14,18,26,24,31,15,11,23)(10,20,28,21,30,27,22,29,16);(2,4)(6,10,11,13,22,24,12,21,14)(7,15,26)(9,18,29,25,31,20,19,23,30)(16,28,27)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 643);