Finite Group Databasebeta

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#72344175

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72210483 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72340864
commutator, cosocle#60045⋊C3C23
FrattiniC22×C42C22×S4
Fitting#72210483D3
socleC23#11713200
hypocenter#2679⋊C3C2×D4
LCSG ▷ #60045⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2

11287 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 11287 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 24

Polycyclic presentation:

2^4,3,2^7|h^2=k,i^2=kl,c^b=cd,e^a=e^2,e^b=e^2,f^a=g,f^b=ghjkl,f^c=fl,f^d=fl,f^e=g,g^a=f,g^b=fhij,g^c=gk,g^d=gk,g^e=fg,h^a=hikl,h^b=hik,h^c=hk,h^e=ik,h^f=hj,h^g=hk,i^a=ikl,i^c=ikl,i^e=hi,i^f=ijl,i^g=ijk,j^e=jk,k^a=l,k^b=l,k^e=kl,l^a=k,l^b=k,l^e=k

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344184

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