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#72342120

order1536
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#6628882 ⋊ C3

Subgroups

subgroupG/H
centerC4AΣL(1,𝔽64)
commutator, cosocle#585C23×C3
FrattiniC22×C4C23×A4
Fitting#6628882C3
socleC2C2×AΣL(1,𝔽64)
hypercenter#41#120⋊C3
hypocenterH(5,𝔽2)C2×C3×D4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

5791 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 5791 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^3,3,2^6|e^2=j,f^2=j,g^2=j,b^a=be,c^a=cj,c^b=cj,e^a=ej,e^b=ej,f^b=fhi,f^d=g,g^b=gh,g^d=fgj,g^f=gj,h^d=ij,h^f=hj,i^d=hij,i^f=ij,i^g=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342145
maximal normal subgroup of#72342145

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