Finite Group Databasebeta

← Prev (#72340930) Next (#72340932) →

#72340931

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#8934620 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72235037
commutator, cosocle, hypocenter#73290C23
Frattini, socleC23C22×#1020
Fitting#8934620D3
LCSG ▷ #73290
UCSC1 ◁ C2

4448 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 4448 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 24 points:

(1,3,11)(2,8,17)(4,16,23,5,14,20)(6,12,21)(7,15,13)(9,19,22,10,18,24);(1,4)(2,10)(3,12)(5,6)(7,9)(11,23)(13,22)(17,24)(18,19)(20,21);(1,2,6,7)(3,11,12,21)(4,9,5,10)(8,17,15,13)(14,20,16,23)(18,24,19,22)

Polycyclic presentation:

2^3,3,2^7|a^2=ijk,e^2=k,f^2=j,g^2=j,h^2=jk,b^a=bijk,d^a=d^2,d^b=dj,e^a=fjk,e^b=ek,e^c=ek,e^d=fk,f^a=ejk,f^b=fj,f^c=fj,f^d=ef,g^a=ghj,g^b=gj,g^d=hj,g^e=gi,g^f=gj,h^b=hjk,h^d=gh,h^e=hik,h^f=hij,i^d=ij,j^a=k,j^d=jk,k^a=j,k^d=j

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340935

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