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#72258697

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#393780 ⋊ D3

Subgroups

subgroupG/H
centerC2#72226168
commutator, cosocleC2×C4×C42⋊C3C23
FrattiniC23×C4C22×S4
Fitting#393780D3
socleC23#85297
hypercenterM16#857⋊C2
hypocenterC42⋊C3#453
LCSG ▷ C2×C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

3511 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 3511 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^6,3,2^4|a^2=e,d^2=f,h^2=j,i^2=k,b^a=bd,c^a=cejk,d^b=df,d^c=df,e^b=ef,g^a=g^2j,g^b=g^2,g^d=gj,g^e=gj,h^a=ijk,h^b=ijk,h^c=hj,h^g=hi,i^a=hjk,i^b=hjk,i^c=ik,i^g=hk,j^a=k,j^b=k,j^g=jk,k^a=j,k^b=j,k^g=j

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258767
maximal normal subgroup of#72258767

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