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#72239408

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239401 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209996
commutator, Fitting, cosocle, hypocenter#72239401C3
FrattiniC22×C43C22⋊A4
socleC23#72209995
LCSG ▷ #72239401
UCSC1 ◁ C2

6903 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 6903 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 64
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 64 points:

(2,6,3)(7,16,10)(8,15,11)(9,14,12)(17,18,23)(19,29,51)(20,34,58)(21,26,41)(22,28,43)(24,44,35)(25,47,55)(27,42,32)(30,52,60)(31,53,40)(33,50,39)(36,45,63)(37,46,64)(38,54,61)(48,59,56)(49,62,57);(2,8,9,7)(3,12)(6,15,14,16)(10,11)(17,55,21,33,22,57,32,56)(18,52,26,29,28,54,27,53)(19,36,38,37,31,20,30,35)(23,64,43,44)(24,50,45,49,46,48,34,25)(39,60,59,61)(40,62,51,47)(41,58,42,63);(1,6)(2,3)(4,16)(5,14)(7,10)(8,11)(9,12)(13,15)(19,30,31,38)(20,37,36,35)(25,48,49,50)(29,54,53,52)(39,62,59,47)(44,63,64,58)

Polycyclic presentation:

3,2^12|b^2=f,c^2=g,d^2=ghkl,e^2=fgjkl,h^2=k,i^2=lm,j^2=m,b^a=cf,c^a=bc,d^a=eghlm,d^b=di,d^c=dh,e^a=de,e^b=ehijlm,e^c=ehiklm,f^a=g,f^d=fm,f^e=fk,g^a=fg,g^d=gk,g^e=gkm,h^a=jm,h^b=hl,h^d=hk,h^e=hkm,i^a=hilm,i^b=il,i^c=il,i^d=im,j^a=hjlm,j^b=jl,j^c=jl,j^d=jkm,j^e=jm,k^a=m,m^a=km

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239411

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", "m");;
gap> coll := SingleCollector(F, [3, 2, 2, 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.7*F.8*F.11*F.12);;
gap> SetPower(coll, 5, F.6*F.7*F.10*F.11*F.12);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12*F.13);;
gap> SetPower(coll, 10, F.13);;
gap> SetConjugate(coll, 2, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.5*F.7*F.8*F.12*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11*F.13);;
gap> SetConjugate(coll, 8, 1, F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11*F.13);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.12*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.8*F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 10, 4, F.10*F.11*F.13);;
gap> SetConjugate(coll, 10, 5, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.13);;
gap> SetConjugate(coll, 13, 1, F.11*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);