Finite Group Databasebeta

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#72341205

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#9061086 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72218857
commutator, cosocle, hypocenter#2736⋊C3C23
Frattini, socleC23C22×#1020
Fitting#9061086D3
LCSG ▷ #2736⋊C3
UCSC1 ◁ C2

3684 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 3684 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 40
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=ij,g^2=ij,h^2=ij,b^a=bi,d^a=dk,d^b=d^2,e^a=ejk,e^b=f,e^d=ef,f^a=fk,f^b=e,f^d=e,g^b=hik,g^c=gk,g^d=h,g^e=gi,g^f=gk,h^b=gijk,h^c=hjk,h^d=ghij,h^e=hjk,h^f=hi,h^g=hij,i^d=ijk,j^d=k,k^b=jk,k^d=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

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