#72271771

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13396 ⋊ C₂
Sylow	C₃ ⋊ #500743

Subgroups

	subgroup	G/H
center	C₂	#10830671
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13396	C₂
socle	C₂×C₃	#24248
hypercenter	#13396	D₃
hypocenter	C₃	#500743
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13396

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=efi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cefhi,c^b=cefgi,d^a=dei,d^b=dei,e^b=ei,e^c=eh,e^d=eh,f^b=fi,f^c=fi,f^d=fi,g^a=gi,g^b=gi,g^c=gi,g^d=gi,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	#72271782
maximal normal subgroup of	#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.5);;
gap> SetPower(coll, 2, F.5*F.6*F.9);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, 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.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, 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, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, 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);