#72220241

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₈
Sylow	C₃⁴ ⋊ C₂²⋊C₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10620192
commutator, socle	C₂×C₃⁴	C₂×C₈
Fitting	C₂²×C₃⁴	C₈
prime radical	C₂²×C₃⋊(C₃⋊(C₃⋊D₃))	C₄
cosocle	C₂×#66472	C₂²
hypercenter	C₂²	#10620192
hypocenter	C₃⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3^4|a^2=c,c^2=e,b^a=bd,f^a=g,f^c=fg^2,f^e=f^2,g^a=fg^2,g^c=f^2g^2,g^e=g^2,h^a=g^2i,h^c=ghi^2,h^e=h^2,i^a=fhi^2,i^c=f^2h^2i^2,i^e=i^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220243, #72220299, #72220302, …
maximal normal subgroup of	#72220243, #72220299, #72220302, …

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