Finite Group Databasebeta

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#72232086

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

Aliases

No aliases for this group.

Splittings

FittingD43 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72224760
commutator, cosocle#59816⋊C3C22
FrattiniC22×C4C2×#1020
FittingD43D3
socleC23#87524
hypocenter#2679⋊C3D4
LCSG ▷ #59816⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2

4210 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 4210 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 — 2 generators on 24 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=cik,c^2=ijk,e^2=i,f^2=ij,g^2=jk,h^2=k,b^a=bc,c^a=cijk,d^a=di,d^b=d^2,d^c=dij,e^a=ei,e^b=fi,e^d=fi,f^a=fij,f^b=eij,f^d=ef,g^a=gj,g^b=hijk,g^c=gi,g^d=hi,g^e=gi,g^f=gik,h^a=hi,h^b=gjk,h^c=hij,h^d=ghijk,h^e=hik,h^f=hij,h^g=hijk,i^b=ij,i^d=ij,j^d=i,k^b=jk,k^d=ik

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72232235, #72340816, #72340829
maximal normal subgroup of#72232235, #72340816, #72340829

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