Finite Group Databasebeta

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#72219696

order15552
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32

Aliases

No aliases for this group.

Splittings

FittingC22×C34 ⋊ C2×SL(2,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC2C2×#72210295
commutator, cosocleC2×#10620211C22×C3
FittingC22×C34C2×SL(2,𝔽3)
socleC2×C34C22×SL(2,𝔽3)
hypercenterC22#72210295
hypocenter#10620211C3×D4
LCSG ▷ C2×#10620211 ▷ #10620211
UCSC1 ◁ C2 ◁ C22

1281 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 1281 conjugacy classes of subgroups — too many to draw. Only its 46 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:

2^2,3,2^4,3^4|a^2=d,e^2=g,f^2=g,b^a=bd,e^c=f,f^c=efg,f^e=fg,h^a=h^2,h^e=hi,h^f=h^2i,h^g=h^2,i^a=i^2,i^c=hi,i^e=hi^2,i^f=hi,i^g=i^2,j^a=h^2ij,j^c=jk,j^e=jk,j^f=k^2,j^g=j^2,k^a=hk,k^e=jk^2,k^f=j,k^g=k^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72219734
maximal normal subgroup of#72219734

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