#72271720

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#29271 ⋊ C₂
Sylow	C₃ ⋊ #500650

Subgroups

	subgroup	G/H
center	C₂	#10830074
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#29271	C₂
socle	C₂×C₃	#24196
hypercenter	#29271	D₃
hypocenter	C₃	#500650
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ #29271

1558 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:

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=fg,c^2=g,d^2=gi,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=cfgh,d^a=di,d^b=de,d^c=di,e^b=eh,e^c=ei,e^d=ehi,f^a=fi,f^b=fi,g^b=gi,h^b=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271743
maximal normal subgroup of	#72271743

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