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#72235031

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72235024 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72235024C3
FrattiniC22×H(3,𝔽4)C22⋊A4
socleC22#72210409
LCSG ▷ #72235024
UCSC1

At least 7766 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has at least 7766 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 64

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

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

Polycyclic presentation:

3,2^12|b^2=l,c^2=m,d^2=klm,e^2=jk,f^2=m,g^2=m,h^2=lm,i^2=lm,b^a=cl,c^a=bc,d^a=eklm,d^b=dg,d^c=di,e^a=de,e^b=ef,e^c=eh,f^a=hik,f^b=fm,f^c=fl,f^d=fj,f^e=fl,g^a=hlm,g^b=gm,g^c=gl,g^d=glm,g^e=gjm,g^f=gm,h^a=fghijkl,h^b=hl,h^c=hlm,h^d=hk,h^e=hl,h^g=hl,i^a=fhlm,i^b=il,i^c=ilm,i^e=iklm,i^f=il,i^h=ilm,j^a=km,j^b=jm,j^c=jl,k^a=jkl,k^b=kl,k^c=klm,l^a=m,m^a=lm

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235040

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