D₄×#6913 #14517828

order	1944
nilpotency class	3
isoclinism	isoclinic class of D₄×#6909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	D₄×#6913
unambiguous semidirect product	C₄⋊(C₂×#6913)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄×C₉ ◁ G

504 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 3 generators on 31 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2222);