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#72266657

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266634 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#72209707
commutator, Fitting, cosocle, hypocenter#72266634C3
FrattiniC23×C4×Q8C22⋊A4
socleC22#72210091
LCSG ▷ #72266634
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:

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

Polycyclic presentation:

3,2^12|b^2=f,c^2=fim,d^2=fijklm,e^2=fgilm,f^2=l,g^2=m,h^2=m,i^2=m,b^a=cgjkl,c^a=bcfgm,c^b=cfgm,d^a=egijkm,d^b=dijk,d^c=dh,e^a=defgjm,e^b=ehjl,e^c=eglm,e^d=efgjm,f^a=fiklm,f^c=fj,f^d=fjk,f^e=fjkm,g^a=gij,g^b=gjlm,g^c=gklm,g^d=gjm,g^e=gkl,h^a=ghjl,h^b=hjk,h^d=hm,h^e=hkl,h^f=hm,h^g=hm,i^a=gjk,i^b=ijk,i^c=ij,i^d=ijkm,i^e=ijlm,i^f=im,i^g=im,i^h=im,j^a=jk,j^c=jm,j^d=jm,k^a=jlm,k^b=km,k^c=km,k^e=km

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266684

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