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#72342068

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#11713201
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#60402D3
prime radical#11710352C2
hypercenterQ8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ Q8

1164 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1164 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:

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

Polycyclic presentation:

2^4,3,2^5|c^2=d,d^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,c^b=cd,d^b=dj,e^c=e^2,f^a=fhj,f^b=fhj,f^c=gj,f^e=fg,g^a=ghij,g^b=ghij,g^c=fj,g^e=f,g^f=gj,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> 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.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
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.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.7*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.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
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);