#72269802

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1312 ⋊ C₂
Sylow	C₃² ⋊ #12062

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#1312	C₂
socle	C₂²×C₃²	C₈×D₄
prime radical	C₂×#3745	C₈
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	#1312	C₃⋊D₃
hypocenter	C₃²	#12062
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1152

Polycyclic presentation:

2^8,3^2|a^2=d,b^2=df,c^2=e,d^2=f,e^2=g,f^2=h,c^a=ce,c^b=cg,e^a=eg,i^a=i^2,j^a=j^2

Conjugacy classes & character table

408 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269811
maximal normal subgroup of	#72269811
maximal quotient of	#72269811

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