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#72342033

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

Aliases

No aliases for this group.

Splittings

Fitting#32947 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713271
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32947D3
prime radical#11714178C2
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

normal subgroups only

This group has 908 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:

(1,16,46,14,23,60,18,42,22,3,20,58,7,43,61,38,6,40,4,37,47,17,45,59)(2,30,56,28,26,57,24,52,33,9,32,64,8,35,62,27,12,55,10,54,50,13,49,63)(5,15,19,36,21,41,44,39)(11,29,31,53,25,34,48,51);(2,8)(3,14)(4,18)(5,21)(6,23)(9,27)(13,28)(15,39)(16,42)(17,38)(22,47)(29,53)(30,54)(31,48)(32,49)(34,51)(35,52)(36,41)(37,43)(40,59)(55,64)(56,62)(57,63)(58,60);(1,2)(3,13)(4,10)(5,12)(6,11)(7,8)(9,17)(14,27)(15,35)(16,34)(18,24)(19,32)(20,31)(21,26)(22,33)(23,25)(28,38)(29,43)(30,41)(36,52)(37,51)(39,54)(40,57)(42,53)(44,49)(45,48)(46,56)(47,50)(55,60)(58,63)(59,64)(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,f^a=g,f^b=fhj,f^e=fg,g^a=f,g^b=ghij,g^e=f,h^a=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, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*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, 5, F.6);;
gap> SetConjugate(coll, 8, 1, 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);