Finite Group Databasebeta

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#72268124

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268112 ⋊ C3
derived#72268108 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC23#35346⋊C3
commutator, cosocle, hypocenter#72268108C2×C3
FrattiniC26C2×C22⋊A4
Fitting#72268112C3
hypercenterC24#85645
LCSG ▷ #72268108
UCSC1 ◁ C23 ◁ C24

7180 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 7180 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 256

Polycyclic presentation:

2,3,2^10|c^2=j,d^2=j,e^2=k,f^2=k,c^a=cgh,c^b=d,d^a=dg,d^b=cdj,d^c=dj,e^b=f,e^c=el,e^d=ei,f^b=efk,f^c=fil,f^d=fl,f^e=fk,g^b=hijl,g^c=gijl,g^d=gl,g^e=gl,h^b=ghijl,h^c=hij,h^d=hijl,h^e=hl,h^f=hl,i^a=il,j^a=jl

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268148
maximal normal subgroup of#72268148

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