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#72227299

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227298 ⋊ C3
derived#8926711 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#35346⋊C3
commutator, cosocle, hypocenter#8926711C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72227298C3
hypercenterC23#85645
LCSG ▷ #8926711
UCSC1 ◁ C22 ◁ C23

4543 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 4543 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 128

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

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

Polycyclic presentation:

2,3,2^9|c^2=k,d^2=k,c^a=cg,c^b=d,d^a=dh,d^b=cdk,d^c=dk,e^b=f,e^c=ei,f^b=ef,f^c=fi,f^d=fi,g^b=h,g^d=gij,g^e=gk,g^f=gk,h^b=ghij,h^c=hij,h^f=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227728, #72227972
maximal normal subgroup of#72227728, #72227972
maximal quotient of#72267936, #72268096, #72268119, …

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