#72270406

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12874 ⋊ C₂
Sylow	C₃ ⋊ #500001

Subgroups

	subgroup	G/H
center	C₂	#10827340
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12874	C₂
socle	C₂×C₃	#23960
hypercenter	#12874	D₃
hypocenter	C₃	#500001
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12874

1828 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:

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

Polycyclic presentation:

2^9,3|b^2=e,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bg,c^a=cgi,c^b=ce,d^b=defi,e^a=ei,e^c=eh,e^d=ehi,f^a=fi,f^c=fi,f^d=fi,g^b=gi,h^c=hi,h^d=hi,j^c=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270453
maximal normal subgroup of	#72270453

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