Finite Group Databasebeta

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#72337583

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C29

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216433
commutator, Frattini, cosocleC2×H(7,𝔽2)C24
prime radical#72227297C2
LCSG ▷ C2×H(7,𝔽2) ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C25 ◁ C2×H(7,𝔽2) ◁ G
LECSG ▷ C2×H(7,𝔽2) ▷ C25 ▷ C22 ▷ C2 ▷ C1

Only the 225 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 225 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:

2^12|d^2=el,c^a=cg,c^b=cf,d^a=dgkl,d^b=dfkl,d^c=de,e^a=eh,e^b=ei,f^a=fj,f^d=fil,g^b=gj,g^d=ghl,g^e=gl,h^b=hk,h^c=hl,h^d=hl,h^f=hl,i^a=ik,i^g=il,j^d=jkl,j^e=jl,k^c=kl,k^d=kl

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

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