Finite Group Databasebeta

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#72258916

order18432
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC28 ⋊ GO+(4,𝔽2)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#72220401C22
Frattini, socleC24#10596770⋊C2
FittingC28GO+(4,𝔽2)
hypocenterC22⋊A42D4
LCSG ▷ #72220401 ▷ C22⋊A42
UCSC1

At least 7633 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has at least 7633 conjugacy classes of subgroups — too many to draw. Only its 9 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:

2^3,3^2,2^8|a^2=c,b^a=bc,d^a=de,d^c=d^2,e^a=de^2,e^b=de^2,e^c=e^2,f^a=i,f^b=im,f^c=fg,f^d=g,f^e=g,g^a=h,g^b=hl,g^d=fg,g^e=fg,h^a=g,h^b=gjk,h^d=hi,i^a=fg,i^b=fj,i^c=hi,i^d=h,j^a=m,j^b=m,j^c=k,j^d=jk,j^e=jk,k^a=lm,k^b=lm,k^c=j,k^d=j,k^e=j,l^a=jk,l^b=jk,l^d=lm,m^a=k,m^b=j,m^c=lm,m^d=l

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258970
maximal normal subgroup of#72258970

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