#72254877

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#10989 ⋊ C₂
Sylow	C₃ ⋊ #133356

Subgroups

	subgroup	G/H
center	C₂²	#69598
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#10989	C₂
socle	C₂²×C₃	#1790
prime radical	#11427646	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	#10989	D₃
hypocenter	C₃	#133356
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #10989

2332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=d,f^2=i,b^a=bf,c^b=cde,d^a=di,d^c=dg,e^a=eh,e^b=eg,e^c=eg,f^c=fh,f^e=fi,h^b=hi,j^c=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#133356, #72254946, #72254983
maximal normal subgroup of	D₃×#133356, #72254946, #72254983

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