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#72242897

order2560
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

FittingC5×#7307 ⋊ C2
SylowC5 ⋊ #144366

Subgroups

subgroupG/H
centerC22#10599758
commutator, cosocleC23×C5×D4C23
FrattiniC23×D4C22×D5
FittingC5×#7307C2
socleC22×C5#2021
hypercenter#7307D5
hypocenterC5#144366
LCSG ▷ C23×C5×D4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #7307

4166 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 4166 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,16,15,14)(18,19)(20,21);(1,3)(2,5,7,6)(4,8)(9,10,15,16)(11,12,14,13)(17,18,20,21,19);(1,2,4,7)(3,5)(6,8)(10,14,11,16)

Polycyclic presentation:

2^9,5|a^2=f,b^2=de,c^2=i,d^2=i,b^a=bd,c^a=cdfgi,c^b=cdei,d^a=dg,d^b=dh,d^c=dg,e^a=eghi,e^b=eh,e^c=eg,e^d=ei,f^b=fgi,f^c=fgi,h^b=hi,h^c=hi,j^c=j^4

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#144366
maximal normal subgroup ofD5×#144366

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