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#72227561

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227556 ⋊ C3
derived#346283 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#346283C2×C3
FrattiniC22×#127C2×A4
Fitting#72227556C3
socleC22#10390⋊C3
LCSG ▷ #346283
UCSC1

642 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 18 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:

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

Polycyclic presentation:

2,3,2^9|a^2=ij,c^2=g,d^2=gk,e^2=j,f^2=jk,b^a=bjk,c^a=cf,c^b=cd,d^a=de,d^b=cj,d^c=dgjk,e^b=f,e^c=egh,e^d=ei,f^b=efghijk,f^c=fik,f^d=fghijk,g^a=gij,g^b=gj,g^e=gj,g^f=gjk,h^a=hij,h^b=hj,h^c=hj,h^d=hk,h^e=hj,h^f=hjk,i^b=ik,i^c=ijk,i^d=ij,j^b=jk,k^b=j

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227562
maximal normal subgroup of#72227562

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