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#72239407

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239396 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#11709219
commutator, Fitting, cosocle, hypocenter#72239396C3
FrattiniC24×C4C22⋊A4
socleC23#2679⋊C3
LCSG ▷ #72239396
UCSC1 ◁ C4

4346 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 4346 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,6)(3,7,12)(4,8,10)(5,9,11)(14,15,18)(19,23,26)(20,24,28)(21,22,27);(1,3,5,4)(6,10,11,12)(15,23,22,24)(18,28,27,26);(1,5)(2,8,9,7)(3,4)(6,10,11,12)(13,18)(14,15)(16,27)(17,26)(19,23)(20,24)(21,22)(25,28)

Polycyclic presentation:

3,2^10|b^2=f,c^2=g,d^2=fghi,e^2=fjk,h^2=k,i^2=k,b^a=bc,c^a=bg,d^a=de,d^b=dh,d^c=di,e^a=dfjk,e^b=ej,e^c=ehjk,f^a=fg,g^a=f,h^a=ik,h^b=hk,i^a=hj,i^c=ik,j^a=hik,j^c=jk

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72260019, #72260812, #72267211
maximal quotient of#72239408, #72239537

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