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#72220482

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

Aliases

No aliases for this group.

Splittings

Fitting, socleC34 ⋊ Q8⋊SL(2,𝔽3)
derivedPSU(3,𝔽2)2 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterPSU(3,𝔽2)2C3
Fitting, socleC34Q8⋊SL(2,𝔽3)
LCSG ▷ PSU(3,𝔽2)2
UCSC1

419 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 21 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:

3,2^6,3^4|b^2=g,c^2=g,d^2=f,e^2=f,b^a=c,c^a=bcg,c^b=cg,d^a=e,d^c=dg,e^a=def,e^b=eg,e^d=ef,h^a=hj^2,h^b=j,h^c=h^2j^2,h^d=j,h^e=h^2j^2,h^f=h^2,h^g=h^2,i^a=ijk^2,i^b=h^2ij,i^c=hij^2,i^d=j^2k,i^e=i^2jk^2,i^f=i^2,i^g=hi,j^b=h^2,j^c=h^2j,j^d=h^2,j^e=h^2j,j^f=j^2,j^g=j^2,k^b=hjk,k^c=hk,k^d=h^2i^2,k^e=h^2i^2k,k^f=k^2,k^g=j^2k

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72220496
maximal normal subgroup of#72220496

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