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#72342143

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

Aliases

No aliases for this group.

Splittings

Fitting#63092 ⋊ D3

Subgroups

subgroupG/H
center, socleC2D4×#1020
commutator, cosocle#5950C23
FrattiniC4C22×#1020
Fitting#63092D3
hypercenterSD16#1020
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ SD16

2805 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 2805 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

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, 1, F.10);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 8, 2, 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);