#72220108

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×#6900 ⋊ C₂²
Sylow	#6900 ⋊ C₂×#120

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#92265⋊C₂
commutator, cosocle	C₂²×C₉²	C₂⁴×C₃
Frattini, socle	C₂²×C₃²	C₂²×C₃×D₃²
Fitting	C₂⁴×#6900	C₂²
hypercenter	C₂⁴	#92265⋊C₂
hypocenter	C₉²	C₂×C₃×#120
LCS	G ▷ C₂²×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₂⁴

11785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^2,3^4|a^2=g,h^3=j,i^3=k,b^a=bf,d^a=dg,h^a=h^2j^2,h^c=h^2j^2,h^e=hj^2,i^a=h^2i,i^c=i^2k^2,i^e=ik^2,j^a=j^2,j^c=j^2,k^a=j^2k,k^c=k^2

Conjugacy classes & character table

324 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220184
maximal normal subgroup of	#72220184

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