#72263258

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72258201 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, cosocle, hypocenter	H(9,𝔽₂)	C₂×C₃
Frattini, socle	C₂	C₂×C₂²⋊(C₂²⋊(C₂²⋊A₄))
Fitting	#72258201	C₃
LCS	G ▷ H(9,𝔽₂)
UCS	C₁ ◁ C₄

Only the 1061 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 1061 normal subgroups are shown; non-normal subgroups are omitted.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^9|a^2=k,c^2=k,d^2=k,e^2=k,f^2=k,c^b=d,d^b=cdk,d^c=dk,e^b=f,e^c=ek,e^d=ek,f^b=efk,f^d=fk,f^e=fk,g^b=j,g^c=gk,g^d=gk,g^f=gk,h^b=i,h^e=hk,h^f=hk,i^b=hi,i^f=ik,j^b=gj,j^d=jk,j^e=jk

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263267, #72265756, #72267211
maximal normal subgroup of	#72265756

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