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#72236028

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72236026 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72235029
commutator, Fitting, cosocle, hypocenter#72236026C3
FrattiniC22×#379A4
socleC23#1304⋊C3
LCSG ▷ #72236026
UCSC1 ◁ C2

1613 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 1613 conjugacy classes of subgroups — too many to draw. Only its 11 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 — 2 generators on 48 points:

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

Polycyclic presentation:

3,2^10|b^2=d,c^2=e,d^2=i,e^2=ik,f^2=ij,b^a=bc,c^a=befgik,c^b=cf,d^a=defhij,d^c=dg,e^a=dghi,e^b=eh,e^d=ek,f^a=fgi,f^b=fgij,f^c=fhij,f^d=fj,f^e=fjk,g^a=ghi,g^b=gj,g^c=gk,h^a=gk,h^b=hk,h^c=hjk,i^a=ijk,j^a=k,k^a=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72238341

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