#72271602

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33638 ⋊ C₂
Sylow	C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831702
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33638	C₂
socle	C₂×C₃	#24338
prime radical	C₂×#5193	C₂²
hypercenter	#33638	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #33638

1426 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)(5,14)(6,21)(8,11)(9,19)(10,25)(12,29)(13,15)(16,17)(18,32)(22,26)(24,33)(28,35)(30,31);(1,2,3)(5,13,16,15)(6,20,22,34)(7,8)(10,11)(12,28)(14,27,17,23)(18,31)(19,25)(21,33,26,24)(29,30);(4,6,19,33,10,26,25,34,9,22,7,24,11,21,8,20)(5,12,27,32,14,18,13,29,16,31,23,35,17,28,15,30);(4,5,8,15,11,17,7,23,9,16,25,13,10,14,19,27)(6,18,20,32,21,12,24,30,22,28,34,35,26,31,33,29)

Polycyclic presentation:

2^9,3|a^2=e,b^2=ehi,c^2=gh,e^2=h,g^2=i,h^2=i,b^a=bg,c^a=ceh,c^b=cegi,d^a=def,d^b=def,d^c=dhi,e^c=eh,e^d=ehi,f^b=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

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500940
maximal normal subgroup of	D₃×#500940

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