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#72214444

order26244
automorphism?
homologyAbelianization (H1) = C22×C9 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72214238 ⋊ C22
derivedC272 ⋊ C22×C9

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC272C22×C9
FrattiniC3×C92C3×D32
Fitting#72214238C22
socleC32#72214420
prime radical#72214404C3
cosocle#14546560C22×C3
LCSG ▷ C272
UCSC1

524 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 53 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:

2^2,3^8|c^3=d,e^3=g,f^3=h,g^3=i,h^3=j,e^a=e^2g^2i^2,e^b=e^2g^2i^2,e^c=eg^2i^2,e^d=ei^2,f^a=e^2f,f^b=f^2h^2j^2,f^c=fh^2j^2,f^d=fj^2,g^a=g^2i^2,g^b=g^2i^2,g^c=gi^2,h^a=g^2h,h^b=h^2j^2,h^c=hj^2,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

holomorph ofD27

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