Finite Group Databasebeta

← Prev (#72226089) Next (#72226091) →

#72226090

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

Aliases

No aliases for this group.

Splittings

FittingC42×M16 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×C4⋊#857
commutator, cosocleC4×C42⋊C3C23
FrattiniC22×C4C22×S4
FittingC42×M16D3
socleC23C2×C4⋊S4
hypercenterM16#857
hypocenterC42⋊C3AGL(1,ℤ/8)
LCSG ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

1087 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 20
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^5,3,2^4|a^2=d,d^2=e,g^2=i,h^2=j,b^a=bd,c^a=ce,c^b=ce,d^b=de,f^a=fi,f^b=f^2,g^b=ghj,g^f=hi,h^b=hj,h^f=gh,i^b=ij,i^f=j,j^f=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226109, #72228331, #72258697
maximal normal subgroup of#72226109, #72228331, #72258697

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