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#72265383

order10240
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72258201 ⋊ D5

Subgroups

subgroupG/H
center, hypercenterC4#72209701
commutator, cosocle, hypocenter#72210028C22
Frattini, socleC2C2×#72209701
Fitting#72258201D5
LCSG ▷ #72210028
UCSC1 ◁ C4

At least 872265 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has at least 872265 conjugacy classes of subgroups — too many to draw. Only its 20 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^2,5,2^9|a^2=l,c^b=c^4,d^b=j,d^c=df,e^b=ij,e^c=deg,f^b=hijk,f^c=ef,g^b=ijk,g^c=f,h^b=fg,h^c=hi,h^d=hl,h^e=hl,h^f=hl,i^b=de,i^c=hik,i^f=il,i^g=il,j^b=d,j^c=ik,j^f=jl,k^b=eg,k^c=jk,k^e=kl,k^f=kl

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265798, #72267347, #72268585, …

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.12);;
gap> SetConjugate(coll, 3, 2, F.3^4);;
gap> SetConjugate(coll, 4, 2, F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 8, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 9, 2, F.4*F.5);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 10, 2, F.4);;
gap> SetConjugate(coll, 10, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.5*F.7);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> SetConjugate(coll, 11, 5, F.11*F.12);;
gap> SetConjugate(coll, 11, 6, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);