Finite Group Databasebeta

← Prev (#72226464) Next (#72226466) →

#72226465

order3200
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

SylowC52 ⋊ #1885

Subgroups

subgroupG/H
centerC2#13104114
commutatorC2×C4×C52C22×C4
FrattiniC2×C4C2×D5⋊AGL(1,𝔽5)
FittingC2×C52×D4C2×C4
socleC2×C52#386
prime radicalC2×D4×C5⋊D5C22
cosocleC2×C4×C5⋊D5C23
hypercenterC2×D4D5⋊AGL(1,𝔽5)
hypocenterC52#1885
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

1058 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 26
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 26 points:

(1,3,5,11)(2,7,6,14)(4,13,12,10)(8,16,15,9)(17,18,20,19)(22,23,26,25);(2,6)(4,12)(7,14)(10,13);(1,2,4,8,5,6,12,15)(3,9,10,14,11,16,13,7)(18,19,20,21)(23,24,26,25)

Polycyclic presentation:

2^7,5^2|a^2=d,c^2=de,d^2=g,e^2=g,b^a=bf,c^a=ce,d^b=dg,d^c=dg,e^b=eg,e^c=eg,f^a=fg,f^c=fg,h^a=h^2,h^c=h^3,h^d=h^4,i^a=h^3i^3,i^c=i^3,i^d=i^4

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226467
maximal normal subgroup of#72226467

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