Finite Group Databasebeta

← Prev (#72260894) Next (#72260896) →

#72260895

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×#8058967 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×AGL(2,ℤ/4)
commutator, cosocle#47814⋊C3C23
FrattiniC24C22×#1020
FittingC2×#8058967D3
socleC23C2×#87524
hypercenterC22AGL(2,ℤ/4)
hypocenter#2844⋊C3C2×D4
LCSG ▷ #47814⋊C3 ▷ #2844⋊C3
UCSC1 ◁ C2 ◁ C22

At least 8294 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has at least 8294 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^7|a^2=j,c^2=d,f^2=j,g^2=j,h^2=jkl,i^2=jk,c^a=cd,c^b=cj,e^c=e^2,f^c=fj,f^e=g,g^c=fg,g^e=fgj,g^f=gj,h^a=hkl,h^c=fgil,h^d=hk,h^e=ikl,h^f=hl,h^g=hjk,i^a=ik,i^c=ghj,i^d=il,i^e=hij,i^f=ijk,i^g=ikl,i^h=ij,k^c=l,k^e=l,l^c=k,l^e=kl

Conjugacy classes & character table

83 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, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10*F.11*F.12);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.11*F.12);;
gap> SetConjugate(coll, 8, 3, F.6*F.7*F.9*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.9*F.11*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.12);;
gap> SetConjugate(coll, 11, 5, F.12);;
gap> SetConjugate(coll, 12, 3, F.11);;
gap> SetConjugate(coll, 12, 5, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);