Finite Group Databasebeta

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#72261304

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C29

Aliases

No aliases for this group.

Splittings

Fitting#72237305 ⋊ D3
derivedC22⋊#5464 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊#5464C23
Frattini, socleC22C22×#87588
Fitting#72237305D3
LCSG ▷ C22⋊#5464
UCSC1

Only the 97 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 97 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:

2^3,3,2^8|g^2=l,h^2=k,b^a=bkl,c^b=ckl,d^a=dl,d^b=d^2,d^c=dkl,e^a=ek,e^b=ef,e^d=f,f^a=fkl,f^d=ef,g^b=hkl,g^d=hl,h^b=gkl,h^d=gh,i^a=ik,i^b=ehjkl,i^c=il,i^d=j,i^g=il,i^h=ikl,j^a=jkl,j^b=efgil,j^c=jk,j^d=ij,j^g=jkl,j^h=jk,k^b=l,k^d=kl,l^b=k,l^d=k

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.11*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.8*F.11*F.12);;
gap> SetConjugate(coll, 7, 4, F.8*F.12);;
gap> SetConjugate(coll, 8, 2, F.7*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.5*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 2, F.5*F.6*F.7*F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
gap> SetConjugate(coll, 12, 2, F.11);;
gap> SetConjugate(coll, 12, 4, F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);