#72247906

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12780 ⋊ C₂
Sylow	C₃ ⋊ #511827

Subgroups

	subgroup	G/H
center	C₂	#10943959
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#12780	C₂
socle	C₂×C₃	#32898
hypercenter	#12780	D₃
hypocenter	C₃	#511827
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #12780

4830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(2,9)(3,12)(5,18)(7,21)(8,25)(13,29)(16,19)(20,24)(34,35);(1,3,6,12)(2,8)(4,13,15,29)(5,17,18,14)(7,20)(9,25)(10,11)(16,31,19,30)(21,24)(22,28)(23,32)(26,27)(33,34,35);(1,2,4,10)(3,11,13,25)(5,7,16,22)(6,9,15,27)(8,12,26,29)(14,28,30,24)(17,32,31,20)(18,21,19,23);(2,7,9,21)(4,15)(8,24,25,20)(10,23,27,22)(11,28,26,32)(13,29)(16,19)(30,31)

Polycyclic presentation:

2^9,3|a^2=h,b^2=e,c^2=hi,f^2=i,g^2=i,b^a=bf,c^a=ch,c^b=cg,d^b=deghi,d^c=dhi,e^a=ei,e^c=eh,e^d=eh,f^c=fhi,f^d=fhi,g^a=ghi,g^b=ghi,g^d=gh,g^e=gi,g^f=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511827
maximal normal subgroup of	D₃×#511827
maximal quotient of	#72254274

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