C₂×C₄×#6873 #14516210

order	1944
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×C₄×#6873

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	H(3,𝔽₃)
commutator	C₃²	C₂×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃×C₉	C₂²×C₃²
socle	C₂²×C₃	C₂×C₃²⋊C₉
prime radical	C₂²×H(3,𝔽₃)	C₂×C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₃×C₄×C₉ ◁ G

208 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 87
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 2 generators on 87 points:

(1,2)(7,8,12,10,15,27,22,33,49,24,14,30,45,32,54,68,57,75,11,16,26,23,34,48,44,58,71)(9,17,35,20,37,60,42,62,79,19,13,25,41,29,47,66,52,70,21,31,53,43,56,74,67,77,85)(18,36,46,39,61,69,64,80,83,40,28,50,65,51,72,82,73,84,38,55,59,63,76,78,81,86,87);(3,4,5,6)(7,9,18)(8,13,28)(10,20,39)(11,21,38)(12,25,46)(14,31,55)(15,29,51)(16,17,36)(19,40,24)(22,42,64)(23,43,63)(26,35,59)(27,47,69)(30,53,50)(32,56,76)(33,52,73)(34,37,61)(41,65,45)(44,67,81)(48,60,78)(49,70,83)(54,74,72)(57,77,86)(58,62,80)(66,82,68)(71,79,87)(75,85,84)

Conjugacy classes & character table

408 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 604);