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#72227319

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×#3573 ⋊ D3

Subgroups

subgroupG/H
centerC4C4⋊#1020
commutator, cosocle#5950C23
FrattiniC4C22×#1020
FittingC2×#3573D3
socleC2C2×C4⋊#1020
hypercenterC2×C8#1020
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

2768 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 2768 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341786
maximal normal subgroup of#72341786

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