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#72267869

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72267865 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72226640
commutator, Fitting, cosocle, hypocenter#72267865C3
FrattiniC25×C4C22⋊A4
socleC23#11709313
LCSG ▷ #72267865
UCSC1 ◁ C2

2774 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 2774 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,7)(3,12,32)(4,16,22)(5,10,24)(6,11,25)(8,23,17)(9,29,15)(13,35,48)(14,34,40)(18,26,46)(19,39,42)(20,33,41)(21,31,43)(27,45,36)(28,44,37)(30,47,38);(3,14)(4,20)(5,21)(8,28,27,35)(13,36)(15,38)(16,39,33,26)(17,37)(22,46)(23,45)(24,43)(29,47)(32,40)(41,42);(1,13,5,17,6,37,21,36)(2,39,31,16,11,26,10,33)(3,19,14,20,15,18,38,4)(7,40,43,32,25,47,24,29)(8,30,35,9,27,34,28,12)(22,44,42,45,41,48,46,23)

Polycyclic presentation:

3,2^11|b^2=f,c^2=fg,d^2=ghijkl,e^2=hil,f^2=l,b^a=bc,c^a=bghk,c^b=cfh,d^a=eghjk,d^b=dfghi,d^c=dghjk,e^a=deil,e^b=ehjk,e^c=efghik,e^d=eil,f^a=fghjl,f^c=fk,f^d=fj,f^e=fkl,g^a=ghkl,g^b=gjl,g^c=gk,h^a=gjk,h^b=hkl,h^c=hjk,i^b=ikl,i^c=ij,j^a=kl,k^a=jk

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267870
maximal normal subgroup of#72267870

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", "l");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.7*F.8*F.9*F.10*F.11*F.12);;
gap> SetPower(coll, 5, F.8*F.9*F.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.7*F.8*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.5*F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);