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#72267533

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267531 ⋊ C3
derived#7642105 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#11710337
commutator, cosocle, hypocenter#7642105C2×C3
FrattiniC22×Q8C2×C22⋊A4
Fitting#72267531C3
hypercenterC23#85645
LCSG ▷ #7642105
UCSC1 ◁ C22 ◁ C23

4441 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 4441 conjugacy classes of subgroups — too many to draw. Only its 41 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)(9,10,15,22)(11,17,23,43,12,44,26,49,27,16,30,38)(13,50,36,20,31,19,33,51,14,18,37,21)(24,67,53,41,54,66,56,70,58,39,61,64)(25,45,62,42,28,71,34,47,60,40,32,72)(29,65,59,48,55,68,35,63,57,46,52,69);(1,8,4,6)(2,3,5,7)(10,50)(13,33)(16,68)(17,67)(18,19)(20,21)(22,51)(24,56)(28,32)(31,37)(38,72)(39,40)(41,42)(43,69)(44,71)(45,46)(47,48)(49,70)(52,55)(54,61)(57,59)(60,62)(63,64)(65,66);(2,5)(6,8)(9,23,15,27)(10,38,22,44)(11,30,26,12)(13,55,33,52)(14,53,36,58)(16,49,43,17)(18,65,20,63)(19,64,21,66)(24,28,56,32)(25,35,34,29)(31,60,37,62)(39,45,41,47)(40,48,42,46)(50,71,51,72)(54,59,61,57)(67,69,70,68)

Polycyclic presentation:

2,3,2^9|a^2=k,c^2=k,d^2=k,e^2=ij,f^2=ij,g^2=k,h^2=k,c^b=cd,d^b=c,d^c=dk,e^a=eg,e^b=f,e^c=ei,f^a=fh,f^b=efij,f^d=fi,f^e=fij,g^a=gk,g^b=h,g^c=gk,g^e=gk,g^f=gik,h^a=hk,h^b=ghik,h^d=hk,h^e=hi,h^f=hk,h^g=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267891
maximal normal subgroup of#72267891
maximal quotient of#72267589, #72267894

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, 1, F.11);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, 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);