Finite Group Databasebeta

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#72341057

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72234656
commutator, cosocle#35346⋊C3C22
FrattiniC24C2×#1020
Fitting#8093108D3
socleC23#87524
prime radical#72247738C2
hypocenterC22⋊#997D4
LCSG ▷ #35346⋊C3 ▷ C22⋊#997
UCSC1 ◁ C2

3550 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 48

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=ck,g^2=k,h^2=k,b^a=bc,d^b=d^2,e^b=f,e^d=f,f^b=e,f^d=ef,g^a=gij,g^b=ghi,g^c=gi,g^d=h,g^e=gk,h^a=hi,h^b=hijk,h^c=hj,h^d=ghk,h^e=hk,h^f=hk,h^g=hk,i^b=ij,i^d=j,j^d=ij

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341062
maximal normal subgroup of#72341062

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