Finite Group Databasebeta

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#72267348

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267327 ⋊ C3
derived#72258228 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#72209713
commutator, cosocle, hypocenter#72258228C2×C3
FrattiniC25×C4C2×C22⋊A4
Fitting#72267327C3
socleC22#72224446
LCSG ▷ #72258228
UCSC1 ◁ C2×C4

Only the 52 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 52 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:

2,3,2^11|e^2=l,f^2=km,i^2=m,j^2=m,k^2=m,l^2=m,c^b=cd,d^b=c,e^a=eh,e^b=ef,e^c=ei,e^d=eilm,f^a=fg,f^b=eilm,f^c=fk,f^d=flm,f^e=fiklm,g^b=h,g^e=gijm,h^b=ghijm,h^f=hijm,i^b=k,i^c=im,i^d=im,i^e=im,i^f=im,j^b=ijkm,j^c=jm,j^d=jm,j^e=jm,j^f=jm,k^b=l,k^c=km,k^e=km,l^b=i,l^d=lm,l^f=lm

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267359

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