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#72267916

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267912 ⋊ C3
derived#355895 ⋊ C2×C3

Subgroups

subgroupG/H
centerC4#10935⋊C3
commutator, cosocle, hypocenter#355895C2×C3
FrattiniC23×#41C2×A4
Fitting#72267912C3
socleC2#72209914
hypercenterC2×C4#1437⋊C3
LCSG ▷ #355895
UCSC1 ◁ C4 ◁ C2×C4

3847 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 3847 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2)(3,59,29,17,18,81)(4,64,68,23,22,115)(5,53,6,30,31,35)(7,15)(8,14)(9,16)(10,69,50,19,41,93)(11,72,101,26,45,119)(12,36,13,34,51,39)(20,99,90,73,65,52)(21,82,27,63,24,80)(25,42,121,85,74,98)(28,62,67,79,40,114)(32,109,37,54,94,57)(33,110,38,55,95,58)(43,104,48,96,46,92)(44,123,49,71,100,91)(47,76,112,86,61,118)(56,66,78,70,60,89)(75,77,127,113,83,108)(84,125,122,105,87,102)(88,103,120,126,124,111)(97,106,128,116,117,107);(1,67,8,68,9,24,7,18)(2,100,15,101,16,46,14,41)(3,29,21,27,22,4,40,28)(5,117,33,118,34,83,32,65)(6,120,38,121,39,84,37,66)(10,50,43,48,45,11,44,49)(12,105,55,98,30,111,54,60)(13,106,58,99,35,77,57,112)(17,47,62,108,64,52,63,107)(19,89,71,87,72,25,96,88)(20,93,75,91,76,26,97,92)(23,102,80,56,81,103,79,42)(31,104,95,119,36,123,94,69)(51,82,110,115,53,114,109,59)(61,78,113,126,73,85,116,125)(70,90,122,127,74,86,124,128)

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=h,e^2=k,f^2=k,g^2=k,h^2=k,i^2=k,j^2=k,c^a=ce,c^b=dgik,d^a=df,d^b=cdi,d^c=di,e^a=ek,e^b=f,e^d=eijk,f^a=fk,f^b=efij,f^c=fijk,f^e=fk,g^a=gk,g^b=hk,g^d=gk,h^a=hk,h^b=ghik,h^c=hk,i^a=ik

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268026, #72287991
maximal quotient of#72268171

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