Finite Group Databasebeta

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#72266627

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266626 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC23#72209707
commutator, Fitting, cosocle, hypocenter#72266626C3
FrattiniC25×Q8C22⋊A4
LCSG ▷ #72266626
UCSC1 ◁ C23

8385 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 8385 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 128

Polycyclic presentation:

3,2^12|b^2=g,c^2=him,d^2=f,e^2=f,g^2=m,h^2=m,i^2=m,j^2=m,b^a=chk,c^a=bcghm,c^b=cghm,d^a=e,d^b=dj,d^c=dhijk,e^a=def,e^b=eik,e^c=eghijm,e^d=ef,f^b=fl,f^c=fk,g^a=hikl,g^c=glm,g^d=glm,g^e=gkl,h^a=gilm,h^b=hlm,h^c=hkl,h^d=hlm,h^e=hkl,i^a=g,i^b=ikl,i^c=ikl,i^d=ikm,i^e=ilm,i^g=im,i^h=im,j^a=ghijkl,j^b=jl,j^c=jklm,j^d=jlm,j^e=jkl,j^i=jm,k^a=klm,k^b=km,l^a=k,l^c=lm

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266628

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