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#72341930

order1536
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#511968 ⋊ C3

Subgroups

subgroupG/H
center, socleC2#32930⋊C3
commutator, cosocle#3573C22×C3
FrattiniC22×C8C22×A4
Fitting#511968C3
prime radical#32948⋊C3C2
hypercenterC16#120⋊C3
hypocenterH(5,𝔽2)C3×D8
LCSG ▷ #3573 ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

388 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^2,3,2^7|b^2=dej,d^2=ej,e^2=j,f^2=j,g^2=j,b^a=bd,d^a=de,e^a=ej,f^a=fhi,f^c=g,g^a=gh,g^c=fgj,g^f=gj,h^c=ij,h^f=hj,i^c=hij,i^f=ij,i^g=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341933
maximal normal subgroup of#72341933

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