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#72341969

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

Aliases

No aliases for this group.

Splittings

Fitting#32948 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713271
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32948D3
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

normal subgroups only

This group has 913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341986
maximal normal subgroup of#72341986

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