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#72266655

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266645 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#72209707
commutator, Fitting, cosocle, hypocenter#72266645C3
FrattiniC25×C4C22⋊A4
socleC23#16297⋊C3
LCSG ▷ #72266645
UCSC1 ◁ C4

3038 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 3038 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 96

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

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

Polycyclic presentation:

3,2^11|b^2=f,c^2=fgh,d^2=fgjk,e^2=fgjl,i^2=l,b^a=bc,c^a=bhj,c^b=cfg,d^a=de,d^b=di,d^c=dghi,e^a=djkl,e^b=egjl,e^c=egijl,e^d=efgk,f^a=fhjkl,f^c=fj,f^d=fjk,f^e=fj,g^a=ghkl,g^b=gj,g^c=gkl,g^d=gjk,g^e=gj,h^a=gjkl,h^b=hkl,h^c=hjkl,h^d=hjl,h^e=hk,i^a=ghij,i^b=ijkl,i^d=ijl,i^e=ik,j^a=kl,k^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72266658

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*F.8);;
gap> SetPower(coll, 4, F.6*F.7*F.10*F.11);;
gap> SetPower(coll, 5, F.6*F.7*F.10*F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.4*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.11*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
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);