Finite Group Databasebeta

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#72253023

order15552
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2×C3A4×((C22⋊C9)⋊C2)⋊C3
commutator, cosocleC3×#63700C2×C32
Frattini, socleC2×C32C3×A4×S4
FittingC33×H(5,𝔽2)C3×D3
prime radicalC3×#13164434C3
hypocenter#63700C2×3−1+2
LCSG ▷ C3×#63700 ▷ #63700
UCSC1 ◁ C2×C3

1019 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 66
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 66 points:

(1,40,23,37,18,48,4,33,6,28,39,47,8,31,26,11,3,41,20,30,27,36,16,44)(2,14,46,7,34,42,12,15,25,24,35,19,13,21,43,22,9,29,10,17,45,5,32,38)(49,52,57,51,55,53,54,50,56)(58,59)(60,66)(62,63)(64,65);(1,29,18,22,6,16,8,42,3,7,27,39)(2,35,45,36,9,47,13,32,25,28,34,44)(4,26,21,5,19,17,20,23,14,24,38,15)(10,40,48,11,30,46,12,31,41,37,33,43)(50,52,55)(53,57,56)(58,64,65,61,62,66,59,60,63)

Polycyclic presentation:

2,3^3,2^2,3^2,2^3|a^2=k,c^3=d,e^2=k,f^2=k,g^3=h,i^2=k,j^2=k,c^a=ch^2,c^b=cd,e^c=f,f^c=efk,f^e=fk,g^a=g^2h^2,g^b=gh,h^a=h^2,i^a=ik,i^g=j,j^a=ij,j^g=ijk,j^i=jk

Conjugacy classes & character table

174 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, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.10);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);