Finite Group Databasebeta

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#72338987

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72338784
commutator#20586C22×C4
Frattini, cosocle#6628901C23
prime radical#72210439C22
LCSG ▷ #20586 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6628901 ◁ G
LECSG ▷ #6628901 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1

5450 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 5450 conjugacy classes of subgroups — too many to draw. Only its 71 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:

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

Polycyclic presentation:

2^12|a^2=f,c^2=dfik,d^2=jk,e^2=i,f^2=k,h^2=l,b^a=bg,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dhjl,e^a=egik,e^b=ei,e^c=eg,e^d=eijk,f^b=fi,f^c=fhjkl,f^d=fjl,f^e=fik,g^a=gi,g^c=gj,g^d=gkl,g^e=gk,g^f=gk,h^a=hj,h^b=hikl,h^c=hj,h^d=hl,h^e=hl,h^g=hl,i^a=ik,i^c=ik,i^d=il,j^b=jkl,j^c=jl,j^e=jl,k^c=kl

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338996, #72339094
maximal normal subgroup of#72338996, #72339094

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