#72214554

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₃²×D₄ ⋊ C₂²
Sylow	C₃² ⋊ #23882

Subgroups

	subgroup	G/H
center	C₂	#11809009
commutator, cosocle	C₂²×C₃²×C₄	C₂⁴
Frattini	C₂²×C₄	C₂²×D₃²
Fitting	C₂³×C₃²×D₄	C₂²
socle	C₂×C₃²	#2403
hypercenter	C₂³×D₄	D₃²
hypocenter	C₃²	#23882
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×D₄

4682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=f,f^2=h,c^a=cg,c^b=ce,d^a=dfh,e^a=eh,f^d=fh,g^b=gh,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72214863

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