#72248486

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2127 ⋊ C₂
Sylow	C₃ ⋊ #147819

Subgroups

	subgroup	G/H
center	C₂²	#70033
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#2127	C₂
socle	C₂²×C₃	#1790
prime radical	#11002541	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#2127	D₃
hypocenter	C₃	#147819
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#2127

2582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4)(6,12)(7,16)(11,19)(13,14)(15,20)(17,18)(22,23);(5,7,8,16)(6,11,12,15)(9,17,10,18)(13,20,14,19)(21,22,23);(2,4)(5,6)(7,15,17,20)(8,13,10,12)(9,14)(11,16)(18,19)

Polycyclic presentation:

2^9,3|a^2=f,b^2=dfh,c^2=defgh,d^2=i,g^2=i,b^a=bd,c^a=cde,c^b=cfgi,d^a=dh,d^b=dh,d^c=dhi,e^a=ei,e^b=ehi,e^c=ehi,e^d=ei,f^b=fhi,f^c=fhi,f^d=fi,g^a=gh,g^b=ghi,g^c=ghi,g^e=gi,g^f=gi,h^a=hi,h^b=hi,j^c=j^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147819, #72248495, #72248627
maximal normal subgroup of	D₃×#147819, #72248495, #72248627
maximal quotient of	#72270451

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