Finite Group Databasebeta

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#72340786

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#19073 ⋊ D3
derived#5464 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#5464C23
FrattiniC42C22×S4
Fitting#19073D3
socleC22C2×#1020⋊C2
LCSG ▷ #5464
UCSC1

985 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 29 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:

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

Polycyclic presentation:

2^3,3,2^6|g^2=i,h^2=j,d^a=d^2,e^a=efhi,e^b=eghj,e^c=eij,e^d=f,f^a=fg,f^b=fg,f^c=fi,f^d=ef,g^a=gi,g^b=gi,g^d=hi,g^e=gj,g^f=gi,h^a=gh,h^b=hj,h^d=gh,h^e=hi,h^f=hij,i^d=j,j^a=ij,j^d=ij

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340795, #72340864
maximal normal subgroup of#72340864
maximal quotient of#72344171

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