Finite Group Databasebeta

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#72209909

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#63040 ⋊ D3
derived#11710846 ⋊ C2

Subgroups

subgroupG/H
center, socleC2#11715850
commutator, cosocle, hypocenter#11710846C2
Frattini, hypercenterC22#87588
Fitting#63040D3
LCSG ▷ #11710846
UCSC1 ◁ C2 ◁ C22

1412 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1412 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:

2,3,2^8|a^2=j,g^2=ij,h^2=ij,b^a=b^2,c^a=cd,c^b=d,d^b=cd,e^a=cefgi,e^b=f,e^c=ei,e^d=ei,f^a=cdfghj,f^b=ef,f^d=fi,g^a=chj,g^b=h,g^c=gi,g^d=gij,g^e=gi,g^f=gj,h^a=cdgj,h^b=ghij,h^c=hj,h^d=hi,h^e=hij,h^f=hi,h^g=hij,i^a=ij

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209911, #72259022
maximal normal subgroup of#72209911
maximal quotient of#72227829, #72228024, #72228061, …

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