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#72238855

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72238851 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209996
commutator, Fitting, cosocle, hypocenter#72238851C3
FrattiniC24×C42C22⋊A4
socleC23#72209995
LCSG ▷ #72238851
UCSC1 ◁ C2

18284 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 18284 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:

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

Polycyclic presentation:

3,2^12|b^2=f,c^2=g,d^2=hilm,e^2=hil,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=el,d^b=dj,d^c=dh,e^a=dehim,e^b=eijkm,e^c=eijlm,e^d=ehim,f^a=gl,g^a=fg,h^a=k,h^b=hm,h^c=hlm,h^d=hlm,h^e=hl,i^a=hikl,i^b=im,i^c=ilm,i^d=ilm,i^e=il,j^a=ijlm,j^b=jm,j^c=jm,j^d=jm,k^a=ik,k^b=klm,k^c=kl,k^d=kl,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#72238857

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.8*F.9*F.12*F.13);;
gap> SetPower(coll, 5, F.8*F.9*F.12);;
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.12);;
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.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10*F.11*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.13);;
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.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);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*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);;
gap> SetConjugate(coll, 10, 1, F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.9*F.11);;
gap> SetConjugate(coll, 11, 2, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
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);