#72247879

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12711 ⋊ C₂
Sylow	C₃ ⋊ #511827

Subgroups

	subgroup	G/H
center	C₂	#10943960
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#12711	C₂
socle	C₂×C₃	#32898
hypercenter	#12711	D₃
hypocenter	C₃	#511827
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ #12711

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=g,f^2=i,g^2=i,b^a=bf,c^a=ch,c^b=ce,d^b=deg,d^c=dhi,e^a=ei,e^b=eh,e^d=eh,f^b=fh,f^d=fhi,g^a=ghi,g^c=gh,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	#72254214

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.7);;
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.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
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, 3, F.7*F.8);;
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);