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#72342684

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#63076 ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC4C22×#1020
commutator, hypocenter#997C22×C4
Fitting#63076D3
socleC2C2×C4×#1020
prime radical#11715827C2
cosocle#5950C23
hypercenterM16#1020
LCSG ▷ #997
UCSC1 ◁ C4 ◁ M16

2768 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 2768 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342708, #72342771, #72342800
maximal normal subgroup of#72342708, #72342771, #72342800

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, 2, 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, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
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, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);