Finite Group Databasebeta

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#72255861

order12288
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72219291 ⋊ D3
derived#72225756 ⋊ C22

Subgroups

subgroupG/H
centerC2#72234286
commutator, cosocle, hypocenter#72225756C22
FrattiniC26C2×#1020
Fitting#72219291D3
socleC25C2×#5934
hypercenterC22#72234220
LCSG ▷ #72225756
UCSC1 ◁ C2 ◁ C22

Only the 40 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 40 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^10|f^2=jl,g^2=lm,c^a=cjm,c^b=c^2,d^b=dejk,d^c=e,e^b=ekm,e^c=de,f^a=fhijk,f^b=defgijkl,f^c=gkm,f^d=fj,f^e=fl,g^a=ghjm,g^b=eghk,g^c=fgjklm,g^d=glm,g^e=gk,g^f=gjklm,h^b=hkm,h^c=i,i^b=hijm,i^c=hi,j^b=jkm,j^c=k,k^c=jkm,l^b=kl,l^c=klm

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

automorphism group ofC2×#95, C2×#120

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10*F.12);;
gap> SetPower(coll, 7, F.12*F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.13);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.11*F.13);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.4*F.5*F.6*F.7*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.7*F.11*F.13);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10*F.13);;
gap> SetConjugate(coll, 7, 2, F.5*F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.12*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.11*F.13);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9*F.10*F.13);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10*F.11*F.13);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 11, 3, F.10*F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.11*F.12);;
gap> SetConjugate(coll, 12, 3, F.11*F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);