Finite Group Databasebeta

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#72231673

order9216
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C25×C3

Aliases

No aliases for this group.

Splittings

FittingC3×#6499874 ⋊ D3
derivedC3×#11710730 ⋊ C22

Subgroups

subgroupG/H
centerC2#72231672
commutator, cosocle, hypocenterC3×#11710730C22
FrattiniC24C2×C3⋊#1020
FittingC3×#6499874D3
socleC2×C3#72209653
hypercenterC22#72231670
LCSG ▷ C3×#11710730
UCSC1 ◁ C2 ◁ C22

7566 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 7566 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,5,8)(3,14,9,6,11,12,10,7,15,13,4,16);(1,3,9,4)(2,6,12,7)(5,10,15,11)(8,13,16,14)(18,19);(17,18,19)

Polycyclic presentation:

2^2,3^2,2^8|b^2=l,c^a=c^2,d^a=d^2,e^a=f,e^d=f,f^a=e,f^d=ef,g^a=h,g^b=gij,g^d=h,g^f=gk,h^a=g,h^b=hi,h^d=gh,h^e=hk,i^a=ij,i^d=j,i^e=il,j^d=ij,j^e=jl,j^f=jl,k^b=kl

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#72209832
maximal normal subgroup ofD3×#72209832

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.12);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);