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#72228050

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72228038 ⋊ C3
derived#432837 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#58990⋊C3
commutator, cosocle, hypocenter#432837C2×C3
FrattiniC23×C4C2×C22⋊A4
Fitting#72228038C3
hypercenterC2×C4#3532⋊C3
LCSG ▷ #432837
UCSC1 ◁ C22 ◁ C2×C4

4277 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 4277 conjugacy classes of subgroups — too many to draw. Only its 38 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:

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

Polycyclic presentation:

2,3,2^9|c^2=j,d^2=j,e^2=gjk,f^2=gijk,g^2=k,h^2=k,b^a=bghik,c^a=cik,c^b=d,d^a=dghik,d^b=cdj,d^c=dj,e^a=ei,e^b=fghik,e^c=eg,e^d=eh,f^a=fghik,f^b=efhijk,f^c=fi,f^d=fgik,f^e=fhijk,g^a=gk,g^b=gik,g^c=gk,g^f=gk,h^a=hk,h^b=gk,h^d=hk,h^e=hk,h^f=hk,i^b=ghi,i^d=ik,i^e=ik,i^f=ik

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228054
maximal normal subgroup of#72228054

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