#72237336

order	5120
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72237327 ⋊ C₅
derived	C₄⁴ ⋊ C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72235877
commutator, hypocenter	C₄⁴	C₄×C₅
Frattini, socle	C₂⁵	C₂×C₂⁴⋊C₅
Fitting	#72237327	C₅
prime radical	C₂×C₄⁴⋊C₅	C₂
cosocle	C₂×C₄⁴	C₂×C₅
LCS	G ▷ C₄⁴
UCS	C₁ ◁ C₂

2238 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 2238 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

transitive degree

μ_t(G) = 40

Polycyclic presentation:

2,5,2^9|a^2=c,d^2=h,e^2=i,f^2=j,g^2=k,d^a=dhijk,d^b=df,e^a=eij,e^b=egk,f^a=fi,f^b=dg,g^a=ghj,g^b=degi,h^b=hj,i^b=ik,j^b=hk,k^b=hik

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237337
maximal normal subgroup of	#72237337

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