Finite Group Databasebeta

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#72253005

order6912
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC26×C3 ⋊ D32

Subgroups

subgroupG/H
center, FrattiniC2C2×#13181790
commutator, cosocleC2×C3×A42C23
FittingC26×C3D32
socleC25×C3C2×D32
hypercenterC22#13181790
hypocenterC3×A42C2×D4
LCSG ▷ C2×C3×A42 ▷ C3×A42
UCSC1 ◁ C2 ◁ C22

5721 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 5721 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 15 points:

(5,6,8,7)(9,12,11,10)(13,14,15);(2,4)(6,7,8);(1,2)(3,4)(11,12)(14,15)

Polycyclic presentation:

2^4,3^3,2^4|b^a=bd,e^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^a=g^2,h^e=hi,h^f=i,i^a=hi,i^c=hi,i^e=h,i^f=hi,j^a=hj,j^c=hik,j^e=hj,j^f=k,k^c=ij,k^e=ik,k^f=jk

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#72215293
maximal normal subgroup ofD3×#72215293

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