Finite Group Databasebeta

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#72258349

order9216
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#6342371 ⋊ C3×D3
derivedC22⋊(C22⋊(C22⋊A4)) ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊(C22⋊(C22⋊A4))C22×C3
Frattini, socleC24C2×#1020⋊C3
Fitting#6342371C3×D3
LCSG ▷ C22⋊(C22⋊(C22⋊A4))
UCSC1

14564 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 14564 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 16

Polycyclic presentation:

2^2,3^2,2^8|d^b=d^2,e^a=eij,e^b=h,e^c=efk,e^d=f,f^a=fi,f^b=g,f^c=ekl,f^d=ef,g^a=gkl,g^b=f,g^c=hi,g^d=h,h^a=hl,h^b=e,h^c=ghj,h^d=gh,i^b=kl,i^c=ij,i^d=j,j^b=k,j^c=i,j^d=ij,k^b=j,k^c=kl,k^d=kl,l^b=ij,l^c=k,l^d=k

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258464
maximal normal subgroup of#72258464

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