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#72218792

order12288
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72216433 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72218512
commutator, cosocle#72209717C23
FrattiniC27C22×S4
Fitting#72216433D3
hypercenterC22#72215886
hypocenter#11710730C2×D4
LCSG ▷ #72209717 ▷ #11710730
UCSC1 ◁ C2 ◁ C22

At least 21284 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has at least 21284 conjugacy classes of subgroups — too many to draw. Only its 42 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:

2^4,3,2^8|f^2=l,g^2=l,c^a=cd,e^b=e^2,f^a=fil,f^b=fikm,f^c=fjm,f^d=fkm,f^e=g,g^a=ghl,g^b=fghijklm,g^c=gjkm,g^d=gjm,g^e=fgl,g^f=gl,h^b=hi,h^c=hj,h^e=hi,h^f=hl,i^c=ik,i^e=h,i^g=il,j^b=jk,j^e=jk,j^f=jm,k^e=j,k^g=km,l^c=lm

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267084
automorphism group of#519

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.13);;
gap> SetConjugate(coll, 6, 4, F.6*F.11*F.13);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.12);;
gap> SetConjugate(coll, 7, 2, F.6*F.7*F.8*F.9*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.13);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.10*F.13);;
gap> SetConjugate(coll, 11, 5, F.10);;
gap> SetConjugate(coll, 11, 7, F.11*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);