Finite Group Databasebeta

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#72209910

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#7514140 ⋊ D3
derived#31076⋊C3 ⋊ C22

Subgroups

subgroupG/H
center, socleC2#72209661
commutator, cosocle, hypocenter#31076⋊C3C22
FrattiniC22×C4C2×#1020
Fitting#7514140D3
hypercenterC4#11715753
LCSG ▷ #31076⋊C3
UCSC1 ◁ C2 ◁ C4

2570 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 2570 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:

2^2,3,2^8|f^2=ij,g^2=h,h^2=k,i^2=k,b^a=bhik,c^a=chik,c^b=c^2,d^a=dk,d^b=dhi,d^c=e,e^b=dehijk,e^c=de,f^a=fjk,f^b=degk,f^c=ghik,f^d=fi,f^e=fj,g^a=ghijk,g^b=efj,g^c=fghj,g^d=gh,g^e=gij,g^f=ghj,h^a=hk,h^b=i,h^c=i,h^d=hk,h^f=hk,i^a=ik,i^b=h,i^c=ijk,i^d=ik,i^e=ik,i^f=ik,i^g=ik,j^b=hijk,j^c=hijk,j^d=jk,j^f=jk

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209912
maximal quotient of#72227873, #72228019, #72228096

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