Finite Group Databasebeta

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#72213221

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72213194 ⋊ C3
derived#72213086 ⋊ C2×C3

Subgroups

subgroupG/H
centerC2#72209714
commutator, cosocle, hypocenter#72213086C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72213194C3
socleC23#11710337
hypercenterC22#72209713
LCSG ▷ #72213086
UCSC1 ◁ C2 ◁ C22

39434 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 39434 conjugacy classes of subgroups — too many to draw. Only its 33 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:

2,3,2^10|a^2=l,c^2=i,d^2=ijk,e^2=l,f^2=l,g^2=l,h^2=l,c^a=ch,c^b=cd,d^a=dg,d^b=cik,d^c=dj,e^a=eh,e^b=ef,e^c=ei,e^d=ejk,f^a=fg,f^b=e,f^c=fk,f^d=fijk,f^e=fl,g^a=gl,g^b=h,g^c=gj,g^e=gjl,g^f=gl,h^a=hl,h^b=ghjl,h^d=hj,h^e=hl,h^f=hj,h^g=hl,i^a=il,i^b=k,j^a=jl,k^a=kl,k^b=ijk

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72213263
maximal quotient of#72267523, #72267629, #72267749

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