Finite Group Databasebeta

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#72223191

order6400
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

SylowC52 ⋊ #7269

Subgroups

subgroupG/H
centerC2#72223190
commutatorC42×C52C42
Frattini, hypercenterC4×C8D5⋊AGL(1,𝔽5)
FittingC4×C52×C8C2×C4
socleC2×C52#1296
prime radicalC22×C5⋊D5#100
cosocleC4×C8×C5⋊D5C22
hypocenterC52#7269
LCSG ▷ C42×C52 ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8

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

normal subgroups only

This group has 756 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

Polycyclic presentation:

2^8,5^2|a^2=d,b^2=c,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bdeg,d^b=df,e^a=eg,e^b=eg,f^b=fh,g^a=gh,g^b=gh,i^a=i^4,i^b=i^3,i^c=i^4,j^a=i^3j,j^b=j^3,j^c=j^4

Conjugacy classes & character table

166 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, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 3, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);