#72248318

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12574 ⋊ C₂
Sylow	C₃ ⋊ #146654

Subgroups

	subgroup	G/H
center	C₂²	#70033
commutator	C₂×C₃×D₄	C₂×C₄²
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#12574	C₂
socle	C₂²×C₃	#1790
prime radical	C₂×#5805	C₂²
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	#12574	D₃
hypocenter	C₃	#146654
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #12574

2434 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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=d,c^2=defhi,e^2=i,g^2=i,b^a=bf,c^a=cfgi,c^b=cfghi,d^a=dh,e^b=ehi,e^c=eh,e^d=ei,f^a=fhi,f^b=fh,f^c=fh,f^d=fi,f^e=fi,g^a=gi,g^b=ghi,g^c=gh,g^d=gi,g^f=gi,h^a=hi,h^b=hi,j^c=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#146654, #72248464, #72248627
maximal normal subgroup of	D₃×#146654, #72248464, #72248627
maximal quotient of	#72253527

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