Finite Group Databasebeta

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#72219852

order15360
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, socleC28 ⋊ ΓL(1,𝔽16)
derived#72218551 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenter#72218551C4
Fitting, socleC28ΓL(1,𝔽16)
prime radical, cosocle#72218553C2
LCSG ▷ #72218551
UCSC1

At least 8260 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has at least 8260 conjugacy classes of subgroups — too many to draw. Only its 11 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:

2^2,3,5,2^8|a^2=b,c^a=c^2,d^a=d^2,d^b=d^4,e^a=eg,e^c=eg,e^d=h,f^a=efgh,f^b=ef,f^c=fgh,f^d=eg,g^c=e,g^d=efh,h^a=egh,h^b=gh,h^c=ef,h^d=fh,i^a=fhl,i^b=ik,i^c=efhi,i^d=ijk,j^a=efi,j^b=l,j^c=ehj,j^d=ij,k^a=egjl,k^c=egk,k^d=ikl,l^a=fik,l^b=j,l^c=eghl,l^d=j

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

automorphism group of#564

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