Finite Group Databasebeta

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#72242397

order6144
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C210

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽4) ⋊ D3
derived#72241426 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209700
commutator, cosocle, hypocenter#72241426C2
FittingH(5,𝔽4)D3
LCSG ▷ #72241426
UCSC1 ◁ C22

82749 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 82749 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 64

Polycyclic presentation:

2,3,2^10|c^2=k,d^2=k,i^2=k,j^2=k,b^a=b^2,c^a=cdek,c^b=d,d^a=def,d^b=cdk,d^c=dk,e^a=f,e^b=f,e^c=ek,e^d=ek,f^a=e,f^b=ef,f^d=fk,g^a=deghjkl,g^b=h,g^e=gl,g^f=gkl,h^a=cefhikl,h^b=gh,h^e=hk,h^f=hl,i^a=deik,i^b=j,i^d=il,i^e=il,i^f=ikl,i^g=ik,i^h=ikl,j^a=cefij,j^b=ijk,j^c=jl,j^e=jk,j^f=jl,j^g=jl,j^h=jk,j^i=jk

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72242443, #72244262

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", "l");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5*F.11);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.4*F.5*F.7*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.11*F.12);;
gap> SetConjugate(coll, 8, 1, F.3*F.5*F.6*F.8*F.9*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.4*F.5*F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 1, F.3*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);