Finite Group Databasebeta

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#72252541

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72238014 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72220729
commutator, cosocleC2×#87547C23
FrattiniC23×D4C22×S4
Fitting#72238014D3
hypercenter#127#1020⋊C2
hypocenter#997B(4,𝔽2)
LCSG ▷ C2×#87547 ▷ C2×#997 ▷ #997
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

At least 7031 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has at least 7031 conjugacy classes of subgroups — too many to draw. Only its 61 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:

2^6,3,2^5|b^2=f,c^2=defl,h^2=l,i^2=l,b^a=bf,c^a=cd,c^b=cde,d^b=df,d^c=df,e^a=ef,e^b=ef,e^c=efl,e^d=el,f^c=fl,g^a=g^2,g^c=g^2,h^a=hijl,h^c=hi,h^g=i,i^a=ijk,i^c=il,i^g=hil,i^h=il,j^a=k,j^c=k,j^g=k,j^h=jl,j^i=jl,k^a=j,k^c=j,k^g=jk,k^i=kl

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267100
maximal normal subgroup of#72267100

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