#72222648

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#334 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#1333

Subgroups

	subgroup	G/H
center	C₂²	#10590593
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂×D₃²
Fitting	C₃²×#334	C₂²
socle	C₂²×C₃²	C₂×#102
prime radical	C₂×#63797	C₄
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	#334	D₃²
hypocenter	C₃²	C₂×#1333
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

1024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=d,b^2=def,d^2=f,e^2=g,f^2=h,g^2=h,b^a=be,d^b=dg,e^a=eh,e^b=eg,f^b=fh,g^b=gh,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222659
maximal normal subgroup of	#72222659
maximal quotient of	#72222649

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