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#72266658

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266649 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#72209707
commutator, Fitting, cosocle, hypocenter#72266649C3
FrattiniC23×C4×Q8C22⋊A4
socleC22#72210091
LCSG ▷ #72266649
UCSC1 ◁ C2×C4

3953 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 3953 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) = 48
transitive degreeμt(G) = 128

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

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

Polycyclic presentation:

3,2^12|b^2=f,c^2=fgh,d^2=fgjkm,e^2=fgklm,f^2=m,g^2=m,h^2=m,i^2=l,b^a=cgjklm,c^a=bcfgm,c^b=cfgm,d^a=ek,d^b=dgil,d^c=dhjkm,e^a=defgjm,e^b=eijklm,e^c=eghijkl,e^d=efgjm,f^a=fghjlm,f^c=fk,f^d=fj,f^e=fjkm,g^a=hk,g^b=gk,g^c=gjl,g^d=gj,g^e=gjkm,h^a=ghjlm,h^b=hjl,h^c=hjkl,h^d=hjklm,h^e=hkl,h^f=hm,h^g=hm,i^a=gikm,i^b=ijkl,i^d=ijklm,i^e=ikl,i^f=im,i^g=im,j^a=kl,j^b=jm,j^c=jm,k^a=jkl,k^c=km

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266677

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