C₃×D₄×#733 #14517822

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	C₃×D₄×#733
unambiguous semidirect product	C₄⋊(C₂×C₃×#733)

Subgroups

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

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

normal subgroups only

This group has 528 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) = 34
transitive degree	μ_t(G) = 324

A minimal faithful permutation representation — 3 generators on 34 points:

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

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, 2216);