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#72239347

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239282 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209988
commutator, Fitting, cosocle, hypocenter#72239282C3
FrattiniC43C22⋊A4
socleC23C22⋊(#95⋊C3)
LCSG ▷ #72239282
UCSC1 ◁ C2

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

normal subgroups only

This group has 920 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

3,2^10|b^2=g,c^2=f,d^2=ij,e^2=k,f^2=j,g^2=i,h^2=jk,b^a=cghj,c^a=bch,c^b=ch,d^a=de,d^b=dik,d^c=dj,e^a=dk,e^b=ei,e^c=eij,f^a=fghik,f^b=fi,g^a=fik,g^c=gik,h^a=hijk,h^b=hij,h^c=hijk,i^a=j,j^a=ik,k^a=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239354, #72239408

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