Finite Group Databasebeta

← Prev (#72267663) Next (#72267665) →

#72267664

order3072
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

derived#11709340 ⋊ C4

Subgroups

subgroupG/H
centerC2#72213648
commutator, hypocenter#11709340C4
FrattiniC25#1020
Fitting#4131221D3
socleC23#85564
prime radical, cosocle#72267652C2
hypercenterC22#11714278
LCSG ▷ #11709340
UCSC1 ◁ C2 ◁ C22

700 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 40
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 40 points:

(1,5,7,3)(2,10)(4,18,16,20)(8,12,14,21)(11,13,15,22)(17,24,19,23)(25,31,28,26,30,37,38,33)(27,39,35,34,36,40,29,32);(1,4,13)(2,8,12)(3,11,18)(5,14,24)(6,16,20)(7,17,21)(9,19,23)(10,15,22)(27,28,29)(31,32,40)(34,39,37)(35,36,38)

Polycyclic presentation:

2^2,3,2^8|a^2=b,d^2=i,e^2=h,f^2=h,g^2=hik,c^a=c^2,d^a=eh,d^b=dik,d^c=eik,e^a=dk,e^b=ehk,e^c=dek,e^d=ek,f^a=fghk,f^c=ghk,f^d=fhj,f^e=fh,g^a=gk,g^c=fgk,g^d=ghij,g^e=gjk,g^f=gk,h^a=i,h^c=hik,i^a=h,i^c=h,j^a=hijk,j^c=hij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267671

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