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#72342084

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

Aliases

No aliases for this group.

Splittings

Fitting#21484 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713272
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#21484D3
prime radical#11714179C2
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

908 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 39 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:

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

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^b=e^2,f^a=fhj,f^b=g,f^e=fg,g^a=ghij,g^b=f,g^e=f,h^b=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, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 8, 2, 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);