Finite Group Databasebeta

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#72209901

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#31077 ⋊ D3
derived#31077⋊C3 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC4#85559
commutator, cosocle, hypocenter#31077⋊C3C2
FrattiniC22×C4#1020
Fitting#31077D3
socleC2#11714279
LCSG ▷ #31077⋊C3
UCSC1 ◁ C4

486 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 11 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:

2,3,2^8|a^2=ij,e^2=g,f^2=gij,h^2=j,b^a=b^2gi,c^a=dj,c^b=d,d^a=c,d^b=cd,e^a=fg,e^b=fgj,e^c=eg,e^d=eghj,f^a=egij,f^b=efj,f^c=fghj,f^d=fgi,f^e=fj,g^a=gij,g^b=gij,g^d=gj,h^a=hij,h^b=ghj,h^c=hj,i^b=g,i^c=ij,i^d=ij

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209903, #72209929
maximal normal subgroup of#72209903
maximal quotient of#72227870, #72228002, #72228047

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