Finite Group Databasebeta

← Prev (#72245240) Next (#72245242) →

#72245241

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#232167 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator#1712⋊C3C2×C4
FrattiniC2×#127C2×S4
Fitting#232167D3
socleC22(#857⋊C2)⋊C4
prime radical#72218248C2
cosocle#19745⋊C3C22
hypocenterC82⋊C3C22⋊C4
LCSG ▷ #1712⋊C3 ▷ C82⋊C3
UCSC1

643 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

(1,7,47,8,39,44,6,26,31,2,14,48)(3,40,16,24,35,15,13,43,4,25,30,10)(5,23,34,18,38,46)(9,12,22,20,28,32)(11,29,37,41,19,36)(17,45,33,27,21,42);(3,11)(7,22)(9,18)(10,15)(12,35)(13,34)(14,32)(21,30)(23,43)(24,46)(25,41)(26,42)(27,29)(28,44)(31,37)(33,39)(36,40)(38,47)(45,48)

Polycyclic presentation:

2^4,3,2^6|b^2=c,f^2=h,g^2=i,h^2=j,i^2=k,b^a=bd,e^a=e^2,f^a=fh,f^b=fi,f^c=fj,f^d=fhj,f^e=ghjk,g^a=fgjk,g^b=ghik,g^c=gk,g^d=gik,g^e=fg,h^a=hj,h^b=hk,h^d=hj,h^e=ij,i^a=hi,i^b=ijk,i^d=ik,i^e=hi,j^e=k,k^a=jk,k^e=jk

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245429
maximal normal subgroup of#72245429

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