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#72341958

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

Aliases

No aliases for this group.

Splittings

Fitting#32895 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713200
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32895D3
hypercenterQ8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ Q8

1368 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1368 conjugacy classes of subgroups — too many to draw. Only its 35 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:

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

Polycyclic presentation:

2^4,3,2^5|c^2=d,d^2=j,b^a=bj,c^a=cj,c^b=cd,d^b=dj,e^a=e^2,e^c=e^2,f^a=g,f^b=fhj,f^c=ghij,f^e=fg,g^a=f,g^b=ghij,g^c=fhj,g^e=f,h^a=hi,h^c=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342104
maximal normal subgroup of#72342104

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