Finite Group Databasebeta

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#72227484

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227479 ⋊ C3
derived#346284 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#346284C2×C3
FrattiniC22×#120C2×A4
Fitting#72227479C3
socleC22#10390⋊C3
LCSG ▷ #346284
UCSC1

1715 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 1715 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,8,12,5,6,2)(3,24,18,27,30,10,14,32,29,15,31,21)(4,22,13,23,16,25,19,9,28,20,26,17)(7,11);(1,17,5,13)(2,28,12,25)(3,4,15,20)(6,24,8,30)(7,32,11,31)(9,10,26,29)(14,18,27,21)(16,19,22,23)

Polycyclic presentation:

2,3,2^9|a^2=i,c^2=g,d^2=g,b^a=bk,c^a=ce,c^b=d,d^a=df,d^b=cdg,d^c=dg,e^a=ej,e^b=f,e^c=ei,e^d=egh,f^a=fjk,f^b=efghik,f^c=fghik,f^d=fi,g^a=gi,g^e=gj,g^f=gjk,h^a=hi,h^b=hjk,h^c=hk,h^d=hj,h^e=hj,h^f=hjk,i^c=ij,i^d=ijk,j^b=jk,k^b=j

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227548
maximal normal subgroup of#72227548
maximal quotient of#72227485, #72268790, #72268809

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