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#72341995

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

Aliases

No aliases for this group.

Splittings

Fitting#63093 ⋊ D3

Subgroups

subgroupG/H
center, socleC2D4×#1020
commutator, cosocle#5950C23
FrattiniC4C22×#1020
Fitting#63093D3
hypercenterQ16#1020
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ Q16

2801 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 2801 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:

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

Polycyclic presentation:

2^4,3,2^5|a^2=j,c^2=dj,d^2=j,b^a=bj,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.

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, 1, F.10);;
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, 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);