Finite Group Databasebeta

← Prev (#72209704) Next (#72209706) →

#72209705

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting#17309 ⋊ D3
derived#11709340 ⋊ C2

Subgroups

subgroupG/H
centerC2#11714280
commutator, cosocle, hypocenter#11709340C2
FrattiniC24#1020
Fitting#17309D3
socleC23#5934
hypercenterC22#85558
LCSG ▷ #11709340
UCSC1 ◁ C2 ◁ C22

426 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 12 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=j,c^2=g,d^2=h,e^2=ghi,f^2=ghi,b^a=b^2,c^a=dg,c^b=dgj,d^a=ch,d^b=cdj,d^c=dj,e^a=dfj,e^b=f,e^c=egj,e^d=ei,f^a=ceghj,f^b=efghi,f^c=fi,f^d=fhj,f^e=fghi,g^a=h,g^b=h,h^a=g,h^b=ghj,i^a=ij,i^b=hij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209709, C2×#72209705, #72260680, …
maximal normal subgroup ofC2×#72209705, #72260680
maximal quotient of#72210085, C2×#72209705, #72266673, …

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