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#72342090

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

Aliases

No aliases for this group.

Splittings

Fitting#60400 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713201
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#60400D3
hypercenterQ8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ Q8

1169 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1169 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:

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

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,e^c=e^2,f^a=fhj,f^b=g,f^c=ghij,f^e=fg,g^a=ghij,g^b=f,g^c=fhj,g^e=f,h^b=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, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, 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, 3, F.7*F.8*F.9*F.10);;
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, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 8, 2, 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);