Finite Group Databasebeta

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#72261866

order6144
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

derivedC22⋊#5464 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC22⋊#5464C2×C4
FrattiniC23C2×#87588
Fitting#72237305D3
socleC22C2×#11715843
prime radical#72260994C2
cosocle#72260975C22
LCSG ▷ C22⋊#5464
UCSC1

Only the 89 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 89 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|a^2=c,g^2=l,h^2=k,b^a=bkl,d^a=d^2,d^b=dl,e^a=ef,e^b=ek,e^d=f,f^b=fkl,f^d=ef,g^a=hl,g^d=hl,h^a=gk,h^d=gh,i^a=ehjkl,i^b=ik,i^c=il,i^d=j,i^g=il,i^h=ikl,j^a=efgil,j^b=jkl,j^c=jk,j^d=ij,j^g=jkl,j^h=jk,k^a=l,k^d=kl,l^a=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, 1, F.3);;
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, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8*F.12);;
gap> SetConjugate(coll, 7, 4, F.8*F.12);;
gap> SetConjugate(coll, 8, 1, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.5*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
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.5*F.6*F.7*F.9*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.11*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, 1, F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
gap> SetConjugate(coll, 12, 1, F.11);;
gap> SetConjugate(coll, 12, 4, F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);