Finite Group Databasebeta

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#72258595

order12288
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#72233781 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72252245
commutator, cosocle#72215518C23
FrattiniC23×C4C22×#1020
Fitting#72233781D3
hypercenterC2×D4#87588⋊C2
hypocenter#11710730C2×D4
LCSG ▷ #72215518 ▷ #11710730
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

At least 41645 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has at least 41645 conjugacy classes of subgroups — too many to draw. Only its 77 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:

2^4,3,2^8|d^2=m,h^2=l,i^2=l,c^a=cd,d^a=dm,d^c=dm,e^b=e^2,f^b=g,f^e=g,g^b=f,g^e=fg,h^b=gi,h^c=hkm,h^e=i,h^g=hl,i^b=fh,i^c=ijm,i^e=hil,i^f=il,i^h=il,j^b=k,j^e=jk,j^f=jm,j^h=jm,j^i=jm,k^b=j,k^e=j,k^g=km,k^h=km,l^c=lm

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

automorphism group ofC22×D8, C22×Q16

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