#72225901

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #141758

Subgroups

	subgroup	G/H
center	C₂²	#72039
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting, prime radical	C₃×#33717	C₂
socle	C₂²×C₃	#1972
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#33717	D₃
hypocenter	C₃	#141758
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #33717

868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=dfhi,c^2=e,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bf,c^a=ce,c^b=cgi,d^b=di,d^c=di,e^a=eg,e^b=ei,f^a=fh,f^b=fi,g^a=gi,h^a=hi,j^a=j^2,j^b=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225906
maximal normal subgroup of	#72225906
maximal quotient of	#72226088, #72226202

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.8*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);