Finite Group Databasebeta

← Prev (#72227828) Next (#72227830) →

#72227829

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, prime radical, cosocle, hypocenter#72227827C2
Frattini, hypercenterC23#87588
Fitting#7664541D3
LCSG ▷ #72227827
UCSC1 ◁ C22 ◁ C23

1948 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 1948 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 32
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2,3,2^9|a^2=i,c^2=k,d^2=k,e^2=i,f^2=i,g^2=k,h^2=k,b^a=b^2,c^a=ck,c^b=d,d^a=cd,d^b=cdk,d^c=dk,e^a=ceik,e^b=f,e^c=ek,f^a=cdef,f^b=efi,f^c=fk,f^d=fk,f^e=fi,g^a=hk,g^b=h,g^e=gj,g^f=gk,h^a=gk,h^b=ghk,h^e=hjk,h^f=hj,h^g=hk,j^a=jk

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227861
maximal normal subgroup of#72227861

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