Finite Group Databasebeta

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#72258804

order9216
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72228377 ⋊ C32
derivedC44 ⋊ C22×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC44C22×C32
Frattini, socleC24C22×A42
Fitting#72228377C32
LCSG ▷ C44
UCSC1

4101 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 4101 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 28
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,5)(3,7,10,8,9,12)(4,6,11)(13,27,20,19,22,16)(14,28,24,26,18,21)(15,25,17);(1,8,4,3)(2,7,6,9)(5,11)(10,12)(14,24,23,20,21,15)(16,22,25,26,27,17)(18,28,19)

Polycyclic presentation:

2^2,3^2,2^8|e^2=i,f^2=j,g^2=k,h^2=l,e^a=eij,e^b=ei,e^c=fi,e^d=fi,f^a=fi,f^b=fj,f^c=ef,f^d=ef,g^a=gij,g^b=gk,g^c=efgij,g^d=hk,h^a=hi,h^b=hl,h^c=ehi,h^d=gh,i^c=j,i^d=j,j^c=ij,j^d=ij,k^c=ijk,k^d=l,l^c=il,l^d=kl

Conjugacy classes & character table

88 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, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.12);;
gap> SetConjugate(coll, 12, 3, F.9*F.12);;
gap> SetConjugate(coll, 12, 4, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);