D₄×#6908 #14517823

order	1944
nilpotency class	3
isoclinism	isoclinic class of D₄×AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	D₄×#6908
unambiguous semidirect product	C₄⋊(C₂×#6908)

Subgroups

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

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

normal subgroups only

This group has 536 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)(13,19,24)(14,18,25)(20,28,23)(22,26,30)(27,31,29);(2,4)(6,10,12)(11,18,20)(19,26,27);(5,7,13,8,14,22,15,23,29)(6,11,19,10,18,26,12,20,27)(9,16,24,17,25,30,21,28,31)

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, 2217);