Finite Group Databasebeta

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#72259006

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72258964 ⋊ C3
derivedH(5,𝔽2)2 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC2#72258874
commutator, cosocle, hypocenterH(5,𝔽2)2C2×C3
FrattiniC2×H(5,𝔽2)C2×C22⋊A4
Fitting#72258964C3
hypercenterC22#72215614
LCSG ▷ H(5,𝔽2)2
UCSC1 ◁ C2 ◁ C22

Only the 44 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 44 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:

2,3,2^10|c^a=cg,c^b=d,d^a=dh,d^b=cd,e^a=ei,e^b=f,e^c=ek,e^d=ek,f^a=fj,f^b=ef,f^d=fk,g^b=h,g^e=gk,h^b=gh,h^e=hk,h^f=hk,i^b=j,i^c=ik,i^d=ik,i^g=il,i^h=il,j^b=ij,j^d=jk,j^h=jl,k^a=kl

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259091
maximal normal subgroup of#72259091

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