Finite Group Databasebeta

← Prev (#72227699) Next (#72227701) →

#72227700

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#72209768C2
Frattini, hypercenterC23#87588
Fitting#8093108D3
LCSG ▷ #72209768
UCSC1 ◁ C22 ◁ C23

3748 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 3748 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 32

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

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

Polycyclic presentation:

2,3,2^9|a^2=ik,c^2=j,d^2=j,b^a=b^2,c^a=deg,c^b=d,d^a=cefgh,d^b=cdj,d^c=dj,e^a=efg,e^b=f,e^c=ejk,e^d=ei,f^a=fgh,f^b=ef,f^c=fijk,f^d=fjk,g^a=h,g^b=h,g^d=gik,h^a=g,h^b=gh,h^c=hik,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72227701

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