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#72340922

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

Aliases

No aliases for this group.

Splittings

Fitting#346283 ⋊ D3
derived#72227559 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72227559C2
FrattiniC25#1020
Fitting#346283D3
socleC22#11714249
LCSG ▷ #72227559
UCSC1

627 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 16 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:

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

Polycyclic presentation:

2,3,2^9|a^2=ijk,c^2=h,d^2=hk,e^2=ghjk,f^2=ghk,b^a=b^2,c^a=cd,c^b=dj,d^a=dhjk,d^b=cdhj,d^c=dhj,e^a=cdefgk,e^b=fjk,e^c=ehi,e^d=eg,f^a=dfj,f^b=efgh,f^c=fghi,f^d=fhijk,f^e=fgh,g^a=gi,g^b=gk,g^c=gk,g^d=gj,g^e=gj,g^f=gjk,h^a=hjk,h^b=hk,h^e=hj,h^f=hjk,i^b=ij,i^c=ij,i^d=ijk,i^e=ijk,i^f=ik,j^a=k,j^b=jk,k^a=j,k^b=j

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340935

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