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#72267720

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowQ8×#1304 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72219808
commutator, Fitting, cosocle, hypocenterQ8×#1304C3
FrattiniC26C22⋊A4
socleC23C22⋊(#95⋊C3)
LCSG ▷ Q8×#1304
UCSC1 ◁ C2

4167 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 4167 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 96

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

3,2^10|b^2=g,c^2=f,d^2=ik,e^2=ik,b^a=bc,c^a=bfhk,c^b=ch,d^a=de,d^b=dj,d^c=di,e^a=d,e^b=ei,e^c=ejk,e^d=eik,f^a=gjk,f^b=fj,g^a=fghj,g^c=gk,h^a=hk,h^b=hk,h^c=hj,i^a=ij,j^a=k,k^a=jk

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267722

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