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#72239533

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239505 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209988
commutator, Fitting, cosocle, hypocenter#72239505C3
FrattiniC43C22⋊A4
socleC23C22⋊(#95⋊C3)
LCSG ▷ #72239505
UCSC1 ◁ C2

1252 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 3 generators on 24 points:

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

Polycyclic presentation:

3,2^10|b^2=g,c^2=f,d^2=ijk,e^2=k,f^2=i,g^2=ik,h^2=jk,b^a=cghjk,c^a=bch,c^b=ch,d^a=eijk,d^b=dijk,d^c=dik,e^a=de,e^b=eik,e^c=ek,f^a=fghi,f^b=fj,g^a=fjk,g^c=gi,h^a=hk,h^b=hijk,h^c=hk,i^a=j,j^a=ik,k^a=ij

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239537

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