#72271419

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13310 ⋊ C₂
Sylow	C₃ ⋊ #500743

Subgroups

	subgroup	G/H
center	C₂	#10830669
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13310	C₂
socle	C₂×C₃	#24248
hypercenter	#13310	D₃
hypocenter	C₃	#500743
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13310

2226 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)(6,16)(7,22)(9,24)(10,27)(11,31)(12,25)(14,33)(15,23)(17,35)(18,34)(19,21)(20,32)(26,28)(29,30);(1,2,3)(6,16)(7,22)(9,24)(10,26)(11,31)(12,25)(14,29)(15,23)(17,35)(18,34)(19,21)(20,32)(27,28)(30,33);(4,6,15,17,9,19,22,34,8,18,7,21,24,35,23,16)(5,10,11,30,12,29,20,27,13,28,32,14,25,33,31,26);(4,5)(6,14,18,30)(7,20)(8,13)(9,25)(10,17,28,21)(11,23)(12,24)(15,31)(16,33,34,29)(19,27,35,26)(22,32)

Polycyclic presentation:

2^9,3|a^2=g,b^2=ef,e^2=h,h^2=i,b^a=be,c^a=cg,c^b=cef,d^b=degi,e^a=eh,e^b=ei,e^c=ehi,e^d=ehi,f^a=fi,f^b=fi,g^b=gi,h^a=hi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271460, #72271782
maximal normal subgroup of	#72271460, #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*F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*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, 1, F.8*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);