Finite Group Databasebeta

← Prev (#72260673) Next (#72260675) →

#72260674

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#2032498 ⋊ D3
derived#11709313 ⋊ C22

Subgroups

subgroupG/H
centerC2#72209661
commutator, cosocle, hypocenter#11709313C22
FrattiniC24C2×#1020
Fitting#2032498D3
socleC23C2×#5934
hypercenterC22#11715753
LCSG ▷ #11709313
UCSC1 ◁ C2 ◁ C22

3164 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 3164 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) = 44
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,10)(3,7,11,9,8,12)(4,6,5)(13,15,18,26)(14,19,22,23)(16,43,28,41,32,42,29,25,31,24,17,40)(20,34,36,37,21,44,35,38,27,33,30,39);(1,9,4,3)(2,11,8,10,6,12,7,5)(13,27,17,35,18,36,32,20)(14,31,21,16,22,28,30,29)(15,39,25,33,26,44,43,37)(19,40,34,41,23,42,38,24)

Polycyclic presentation:

2^2,3,2^8|a^2=k,b^2=k,d^2=h,e^2=hi,b^a=bk,c^a=c^2,c^b=chk,d^a=dehik,d^b=dhk,d^c=ehk,e^a=ehi,e^b=ehik,e^c=dek,e^d=ek,f^c=g,f^d=fhij,f^e=fhi,g^a=fg,g^c=fg,g^d=gij,g^e=gjk,h^a=ik,h^c=hi,i^a=hk,i^c=hk,j^a=hij,j^c=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261057

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