#72270626

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#23895 ⋊ C₂
Sylow	C₃ ⋊ #72219157

Subgroups

	subgroup	G/H
center	C₂²	#10974246
commutator, cosocle	C₂⁵×C₃	C₂⁵
Frattini	C₂⁵	C₂⁴×D₃
Fitting	C₂×C₃×#23895	C₂
socle	C₂²×C₃	#35346
hypercenter	C₂×#23895	D₃
hypocenter	C₃	#72219157
LCS	G ▷ C₂⁵×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×#23895

At least 6512 conjugacy classes of subgroups; only the 1371 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|a^2=i,c^2=h,d^2=j,b^a=bg,c^a=cf,d^a=di,d^b=dh,e^a=ei,e^c=eh,e^d=ej,f^a=fj,f^c=fj,f^d=fj,g^d=gj,h^a=hj,i^c=ij,k^b=k^2

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270642, D₃×#72219157, #72271657
maximal normal subgroup of	#72270642, D₃×#72219157, #72271657

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