#72271721

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13374 ⋊ C₂
Sylow	C₃ ⋊ #500003

Subgroups

	subgroup	G/H
center	C₂	#10827339
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13374	C₂
socle	C₂×C₃	#23960
hypercenter	#13374	D₃
hypocenter	C₃	#500003
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13374

1840 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)(4,7,10,23)(5,13,12,29)(6,18)(8,24,22,9)(11,26)(14,31,30,15)(16,28)(17,34)(20,21)(25,32);(1,2,3)(5,12)(6,17,20,34)(7,8)(9,24)(11,25,16,32)(13,30)(14,29)(18,35,21,19)(22,23)(26,33,28,27);(4,6,8,19,9,21,7,17,10,20,22,35,24,18,23,34)(5,11,14,27,15,28,13,25,12,16,30,33,31,26,29,32);(4,5)(6,16,20,11)(7,13)(8,14)(9,15)(10,12)(17,32,34,25)(18,28,21,26)(19,33,35,27)(22,30)(23,29)(24,31)

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,c^2=g,d^2=gi,e^2=h,h^2=i,b^a=bf,c^a=cgi,c^b=cefg,d^b=deg,d^c=dhi,e^c=eh,e^d=eh,f^a=fi,g^b=gi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271743, #72271782
maximal normal subgroup of	#72271743, #72271782

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