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#72220495

order3456
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

FittingC32×Q8 ⋊ GL(2,𝔽3)
derivedQ8⋊(PSU(3,𝔽2)⋊C3) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#13181783
commutator, cosocle, hypocenterQ8⋊(PSU(3,𝔽2)⋊C3)C2
FittingC32×Q8GL(2,𝔽3)
socleC2×C32#5931
LCSG ▷ Q8⋊(PSU(3,𝔽2)⋊C3)
UCSC1 ◁ C2

303 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 16 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:

2,3,2^6,3^2|c^2=h,d^2=h,e^2=g,f^2=g,b^a=b^2,c^a=cd,c^b=d,d^a=dh,d^b=cdh,d^c=dh,e^a=ef,e^b=f,e^d=eh,f^a=fg,f^b=efg,f^c=fh,f^e=fg,i^a=ij^2,i^b=ij,i^c=j^2,i^d=i^2j,i^e=j^2,i^f=i^2j,i^g=i^2,i^h=i^2,j^a=j^2,j^c=i,j^d=ij,j^e=i,j^f=ij,j^g=j^2,j^h=j^2

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72220496, #72244547
maximal quotient 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");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 9, 1, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.10^2);;
gap> SetConjugate(coll, 9, 4, F.9^2*F.10);;
gap> SetConjugate(coll, 9, 5, F.10^2);;
gap> SetConjugate(coll, 9, 6, F.9^2*F.10);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.9);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);