#72271513

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #128059

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4527
commutator	C₂²×C₃×C₄	C₂×C₄²
Frattini	C₂³×Q₈	C₂²×D₃
Fitting	C₃×#20591	C₂
socle	C₂²×C₃	C₂×#342
prime radical	C₂³×C₃×#41	C₂²
cosocle	C₂³×C₃×Q₈	C₂³
hypercenter	#20591	D₃
hypocenter	C₃	#128059
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ #20591

3052 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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=dfi,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ci,c^b=ch,d^a=dg,d^b=dgh,d^c=di,e^b=egi,e^d=ei,f^a=fhi,f^b=fgh,f^c=fi,f^e=fi,g^a=gi,h^a=hi,j^b=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271516, #72271537, #72271617
maximal normal subgroup of	#72271516, #72271537, #72271617

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