#72247806

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12713 ⋊ C₂
Sylow	C₃ ⋊ #511827

Subgroups

	subgroup	G/H
center	C₂	#10943960
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#12713	C₂
socle	C₂×C₃	#32898
hypercenter	#12713	D₃
hypocenter	C₃	#511827
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ #12713

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:

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=efi,c^2=h,f^2=i,g^2=i,b^a=be,c^a=ci,c^b=cg,d^b=defghi,d^c=dh,e^a=ei,e^b=eh,e^c=ei,e^d=eh,f^a=fhi,f^b=fh,f^c=fh,f^d=fi,f^e=fi,g^a=gi,g^b=gh,g^d=ghi,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	#72248142

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