Finite Group Databasebeta

← Prev (#72210003) Next (#72210005) →

#72210004

order1536
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#181415 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#181415C3
FrattiniC2×C42C22⋊A4
socleC22C22⋊(#95⋊C3)
LCSG ▷ #181415
UCSC1

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

normal subgroups only

This group has 535 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:

3,2^9|b^2=f,c^2=g,d^2=gi,e^2=fgh,f^2=i,g^2=j,b^a=cfhi,c^a=bch,c^b=ch,d^a=eghi,d^b=dhj,d^c=di,e^a=dehj,e^b=ehij,e^c=ehi,e^d=ehj,f^a=gj,f^c=fi,f^d=fi,f^e=fi,g^a=fghi,g^b=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hj,h^d=hj,h^e=hij,i^a=j,j^a=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210010, #72235726, C2×#72210004
maximal normal subgroup of#72235726, C2×#72210004
maximal quotient ofC2×#72210004, #72240072

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