#72243529

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14552192 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃×#4169⋊C₃
commutator, cosocle	C₃²×C₉	C₂×C₃³
Frattini, socle	C₃³	C₃³×D₃
Fitting	#14552192	C₂
prime radical	C₃²×C₃⋊AGL(1,ℤ/9)	C₃
hypercenter	C₃²	#4169⋊C₃
hypocenter	C₃×C₉	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃ ◁ C₃²

1780 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 27
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2,3^7|b^3=e,f^3=h,c^b=ce,f^a=f^2h^2,f^b=fg,f^d=fh^2,g^a=g^2,h^a=h^2

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72243535

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);