#72243765

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#532 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#3151

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#3811
commutator, cosocle	C₂×C₃²×C₄	C₂⁵
Frattini	C₂×C₄	C₂³×D₃²
Fitting	C₃²×#532	C₂²
socle	C₂²×C₃²	C₂³×D₄
hypercenter	#532	D₃²
hypocenter	C₃²	C₂×#3151
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

8764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^8,3^2|c^2=fh,d^2=fh,e^2=g,f^2=h,b^a=bh,c^a=cf,c^b=ch,d^a=df,d^b=dh,e^a=eg,f^a=fh,i^d=i^2,j^c=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×#3151
maximal normal subgroup of	D₃²×#3151

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