Finite Group Databasebeta

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#72241545

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

Aliases

No aliases for this group.

Splittings

FittingC24×C32 ⋊ D3×D4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocleC3⋊D3×C22⋊A4C23
FrattiniC22S4×GO+(4,𝔽2)
FittingC24×C32D3×D4
socleC22×C32D4×S4
hypocenterC32×C22⋊A4C2×D4
LCSG ▷ C3⋊D3×C22⋊A4 ▷ C32×C22⋊A4
UCSC1

4594 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 4594 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 14
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 14 points:

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

Polycyclic presentation:

2^4,3^3,2^4|c^b=cd,e^a=e^2,f^b=f^2,f^c=f^2g^2,f^d=f^2,g^b=f^2g,g^d=g^2,h^a=i,h^b=hj,h^e=hi,i^a=h,i^b=ijk,i^e=h,j^a=jk,j^e=k,k^e=jk

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

automorphism group ofC22×D32

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