#72248347

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#23897 ⋊ C₂
Sylow	C₃ ⋊ #72213329

Subgroups

	subgroup	G/H
center	C₂²	#10943957
commutator, cosocle	C₂³×C₃×D₄	C₂⁴
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₂×C₃×#23897	C₂
socle	C₂²×C₃	#32898
hypercenter	C₂×#23897	D₃
hypocenter	C₃	#72213329
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ C₂×#23897

At least 10686 conjugacy classes of subgroups; only the 503 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|a^2=ij,b^2=f,h^2=j,b^a=bh,c^b=cf,d^b=dfg,d^c=de,f^a=fj,g^a=gi,h^c=hj,h^d=hi,h^g=hj,i^b=ij,k^d=k^2

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213329
maximal normal subgroup of	D₃×#72213329
maximal quotient of	#72271470

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 4, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);