C₃×D₄×C₂₇⋊C₃ #14517817

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

Aliases

direct product	C₃×D₄×C₂₇⋊C₃
unambiguous semidirect product	C₄⋊(C₂×C₃×C₂₇⋊C₃)
unambiguous semidirect product	(D₄×C₂₇)⋊C₃²

Subgroups

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

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

normal subgroups only

This group has 480 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) = 34
transitive degree	μ_t(G) = 324

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

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

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, 2211);