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#72341128

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#9061103 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72258118
commutator, cosocle#58990⋊C3C22
FrattiniC24C2×#1020
Fitting#9061103D3
socleC23#87524
hypocenter#2679⋊C3D4
LCSG ▷ #58990⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2

2750 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 2750 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=k,g^2=ij,h^2=ijk,b^a=bc,d^a=djk,d^b=d^2j,d^c=dj,e^a=ejk,e^b=fj,e^d=f,f^a=fj,f^b=ejk,f^d=ef,g^a=gj,g^b=eghij,g^c=gj,g^d=hjk,g^e=gi,g^f=gj,h^a=hk,h^b=efhjk,h^c=hk,h^d=ghik,h^e=hijk,h^f=hij,h^g=hik,i^d=ij,j^b=jk,j^d=k,k^d=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341182
maximal normal subgroup of#72341182

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