Finite Group Databasebeta

← Prev (#72344171) Next (#72344173) →

#72344172

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

Aliases

No aliases for this group.

Splittings

Fitting#177861 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72340796
commutator, cosocle#60045⋊C3C22
FrattiniC22×C42C2×S4
Fitting#177861D3
socleC23#73081
hypocenter#2679⋊C3D4
LCSG ▷ #60045⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2

1526 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 19 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 — 2 generators on 24 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=c,g^2=j,h^2=jk,b^a=bc,d^a=d^2,d^b=d^2,e^a=fj,e^b=fgijk,e^c=ek,e^d=f,f^a=ek,f^b=eghi,f^c=fj,f^d=ef,g^a=ghj,g^b=ghjk,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^b=k,j^d=jk,k^a=j,k^b=j,k^d=j

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344175
maximal normal subgroup of#72344175

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