#72248289

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12690 ⋊ C₂
Sylow	C₃ ⋊ #511793

Subgroups

	subgroup	G/H
center	C₂	#10943960
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12690	C₂
socle	C₂×C₃	#32898
hypercenter	#12690	D₃
hypocenter	C₃	#511793
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #12690

4952 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)(4,16)(12,28)(13,14)(17,18)(20,32)(23,31)(26,27)(33,34,35);(1,4,6,16)(2,8,9,24)(3,12,15,28)(5,18,19,17)(7,20,22,32)(10,23,25,31)(11,26,21,27)(13,29,14,30);(1,3,6,15)(2,7)(4,12,16,28)(5,14,19,13)(8,20)(9,22)(10,11)(17,30,18,29)(21,25)(23,27)(24,32)(26,31)(34,35);(1,2,5,10,6,9,19,25)(3,11,13,7,15,21,14,22)(4,8,17,23,16,24,18,31)(12,27,29,20,28,26,30,32)

Polycyclic presentation:

2^9,3|a^2=e,b^2=eghi,c^2=i,e^2=i,g^2=i,b^a=bg,c^a=cf,c^b=cf,d^a=defghi,d^b=defghi,d^c=di,e^b=ei,e^c=eh,e^d=ehi,f^a=fh,f^e=fi,g^c=gh,g^d=ghi,g^f=gi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511793
maximal normal subgroup of	D₃×#511793
maximal quotient of	#72248599

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