#72220362

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#6342371 ⋊ D₅
derived	C₂⁴⋊(C₂⁴⋊C₅) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴⋊(C₂⁴⋊C₅)	C₂²
Frattini, socle	C₂⁴	C₂×(C₂⁴⋊C₅)⋊C₂
Fitting	#6342371	D₅
LCS	G ▷ C₂⁴⋊(C₂⁴⋊C₅)
UCS	C₁

27620 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 27620 conjugacy classes of subgroups — too many to draw. Only its 10 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:

2^2,5,2^8|c^b=c^4,d^a=dj,d^b=de,d^c=de,e^a=ehi,e^c=df,f^a=fhjk,f^b=eg,f^c=dg,g^a=gij,g^b=ef,g^c=d,h^b=ik,h^c=ik,i^b=hk,i^c=hi,j^b=hij,j^c=hij,k^c=hijk

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72220408

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