Finite Group Databasebeta

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#72260440

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC24×#392 ⋊ D3

Subgroups

subgroupG/H
centerC22C2×#11715690
commutator, cosocleC2×C22⋊(#95⋊C3)C23
FrattiniC24C22×#1020
FittingC24×#392D3
socleC26C22×S4
hypercenterC23#11715690
hypocenterC22⋊(#95⋊C3)C2×D4
LCSG ▷ C2×C22⋊(#95⋊C3) ▷ C22⋊(#95⋊C3)
UCSC1 ◁ C22 ◁ C23

At least 12210 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has at least 12210 conjugacy classes of subgroups — too many to draw. Only its 89 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:

2^4,3,2^7|a^2=l,c^2=d,f^2=j,g^2=k,h^2=j,i^2=k,c^a=cd,c^b=cj,e^b=ekl,e^c=e^2,f^a=fjl,f^c=fl,f^e=gjl,g^a=gkl,g^c=fgjk,g^e=fgl,g^f=gl,h^a=hjl,h^c=fgijl,h^e=ijl,h^g=hl,i^a=ikl,i^c=ghj,i^e=hil,i^f=il,i^h=il,j^e=k,k^c=jkl,k^e=jkl

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

automorphism group of#1273, #1455

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