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#72342154

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#11713272
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#21483D3
prime radical#3025⋊C3C22
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

normal subgroups only

This group has 901 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

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