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#72266616

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266613 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22#72209707
commutator, Fitting, cosocle, hypocenter#72266613C3
FrattiniC27C22⋊A4
socleC24#73292
LCSG ▷ #72266613
UCSC1 ◁ C22

5266 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 5266 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) = 48
transitive degreeμt(G) = 192

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

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

Polycyclic presentation:

3,2^11|b^2=i,c^2=h,d^2=f,e^2=f,b^a=bc,c^a=bghk,c^b=cg,d^a=de,d^b=dghj,d^c=dghik,e^a=d,e^b=egjk,e^c=egij,e^d=ef,f^b=fk,f^c=fkl,g^a=gk,g^b=gk,g^c=gkl,h^a=il,h^b=hkl,h^d=hl,h^e=hkl,i^a=ghikl,i^c=ik,i^d=ik,i^e=il,j^a=gij,j^b=jkl,j^c=jk,j^d=jkl,j^e=jk,k^a=l,l^a=kl

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72266627

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