Finite Group Databasebeta

← Prev (#72245980) Next (#72245982) →

#72245981

order3456
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C22×((C22⋊C9)⋊C2)⋊C3
commutator, hypocenterQ8⋊C9C22×C3×C4
FrattiniC3×C4C22×C3×S4
Fitting, prime radicalC32×#585D3
socleC2×C3C22×C3×SL(2,ℤ/4)
cosocle#41⋊C9C23×C3
hypercenter#41((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C4 ◁ #41

1259 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 153 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) = 57
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 57 points:

(1,15,17,6,33,38)(2,43,28)(3,19,36,14,31,18)(4,20,35,16,5,39)(7,41,45)(8,27,44)(9,10,46)(11,12,37,13,34,32)(21,30,29)(22,40,47)(23,42,24)(25,26,48)(49,53,54,51,57,56,55,50,52);(1,3,4,11,6,14,16,13)(2,7,9,22,8,21,25,24)(5,17,15,35,20,38,33,39)(10,28,27,46,26,44,43,48)(12,32,31,36,34,37,19,18)(23,42,41,45,40,47,30,29)(50,52)(51,55)(53,56)(54,57);(1,2,6,8)(3,7,14,21)(4,9,16,25)(5,10,20,26)(11,22,13,24)(12,23,34,40)(15,27,33,43)(17,28,38,44)(18,29,36,45)(19,30,31,41)(32,42,37,47)(35,46,39,48)

Polycyclic presentation:

2^3,3,2,3^2,2^3|a^2=j,b^2=e,e^2=j,f^3=g,h^2=j,i^2=j,c^a=cj,f^b=f^2g^2,f^d=fg,g^b=g^2,h^b=hj,h^f=i,i^b=hi,i^f=hij,i^h=ij

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72253040
maximal normal subgroup of#72253040

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