#72214310

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#493 ⋊ C₂²
Sylow	C₃² ⋊ #61110

Subgroups

	subgroup	G/H
center	C₂	#11902357
commutator, cosocle	C₂×C₃²×C₄	C₂⁵
Frattini	C₂×C₄	C₂³×D₃²
Fitting	C₃²×#493	C₂²
socle	C₂×C₃²	C₂×D₄²
hypercenter	#493	D₃²
hypocenter	C₃²	#61110
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

5696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|b^2=g,c^2=f,g^2=h,b^a=bg,c^a=cgh,d^b=dgh,d^c=df,e^b=eh,f^a=fh,g^a=gh,g^d=gh,i^a=i^2,i^d=i^2,i^e=i^2,j^a=i^2j,j^d=j^2,j^e=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219759, #72219777, #72219786
maximal normal subgroup of	#72219759, #72219777, #72219786

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