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#72239995

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239981 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72210010
commutator, Fitting, cosocle, hypocenter#72239981C3
FrattiniC24×C42C22⋊A4
socleC23#72210009
LCSG ▷ #72239981
UCSC1 ◁ C2

15848 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 15848 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

3,2^12|b^2=f,c^2=g,d^2=ghilm,e^2=fghilm,f^2=l,g^2=lm,h^2=lm,i^2=lm,j^2=m,k^2=m,b^a=cfm,c^a=bc,d^a=eglm,d^b=dj,d^c=dh,e^a=dehil,e^b=eim,e^c=eikm,e^d=ehil,f^a=gl,g^a=fg,h^a=k,h^b=hlm,h^c=hlm,h^d=hlm,h^e=hlm,i^a=hik,i^b=ilm,i^c=ilm,i^d=ilm,i^e=ilm,j^a=ikm,j^b=jm,j^c=jlm,j^d=jm,j^e=jlm,k^a=ijl,k^c=km,k^e=km,l^a=lm,m^a=l

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239999

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