#72222656

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1313 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#12068

Subgroups

	subgroup	G/H
center	C₄	#11860162
commutator	C₃²×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂²×D₃²
Fitting	C₃²×#1313	C₂²
socle	C₂×C₃²	C₂×#1580
prime radical	#11899513	C₄
cosocle	C₃²×C₄×C₈	C₂⁴
hypercenter	#1313	D₃²
hypocenter	C₃²	C₂×#12068
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

2031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3^2|a^2=e,b^2=efgh,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bf,d^a=dfi,d^b=dh,e^b=ei,e^d=ei,f^a=fh,f^b=fi,h^a=hi,j^a=j^2,j^c=j^2,k^a=j^2k,k^c=k^2

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222660
maximal normal subgroup of	#72222660

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 3, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);