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#72235873

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72235753 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72235273
commutator, Fitting, cosocle, hypocenter#72235753C3
FrattiniC26C22⋊A4
socleC23#73292
LCSG ▷ #72235753
UCSC1 ◁ C2

980 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 72
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 72 points:

(2,3,7)(5,6,8)(10,11,17)(12,13,29)(14,15,16)(18,42,68)(19,46,60)(20,25,41)(21,35,39)(22,26,40)(23,51,63)(24,56,44)(27,59,72)(28,57,61)(30,36,67)(31,34,66)(32,33,65)(37,49,70)(38,47,55)(43,54,69)(45,58,71)(48,53,52)(50,64,62);(2,6)(3,8)(9,67)(10,61)(11,53,26,64)(12,34,32,13)(14,66)(15,29)(16,65)(17,72,41,68)(18,19,45,49)(20,43)(21,63)(22,47)(23,24,28,42)(25,70,35,46)(27,50,44,48)(30,36,31,33)(37,69,60,51)(38,52,57,62)(39,71,40,56)(54,59,55,58);(1,5,4,7)(2,8,6,3)(9,39)(10,25,22,11)(12,60,31,37)(13,51)(14,41)(15,17)(16,40)(18,55,44,28)(19,42,49,59)(20,26,21,35)(23,27,54,45)(24,48,58,50)(29,71,65,68)(30,62,32,52)(33,38)(34,57)(36,69)(43,53,47,64)(46,63,70,61)(56,67,72,66)

Polycyclic presentation:

3,2^10|b^2=f,c^2=h,d^2=ghij,e^2=ik,b^a=bc,c^a=bghk,c^b=cg,d^a=de,d^b=dfhi,d^c=dgijk,e^a=dfg,e^b=efk,e^c=eghik,e^d=efgi,f^a=fghj,f^c=fk,f^d=fj,g^a=gk,g^b=gk,g^c=gj,g^d=gk,g^e=gj,h^a=fjk,h^b=hj,h^d=hjk,h^e=hjk,i^a=ghij,i^c=ik,i^d=ij,j^a=k,k^a=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72235874

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