#72218553

order	7680
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁸ ⋊ C₃×D₅
derived	C₂⁴⋊(C₂⁴⋊C₅) ⋊ C₂×C₃
Hall	#72209701 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴⋊(C₂⁴⋊C₅)	C₂×C₃
Fitting, socle	C₂⁸	C₃×D₅
LCS	G ▷ C₂⁴⋊(C₂⁴⋊C₅)
UCS	C₁

15804 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 15804 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:

2,3,5,2^8|c^a=c^4,d^a=df,d^b=eg,d^c=df,e^a=defg,e^b=fg,e^c=eg,f^b=dg,f^c=e,g^a=fg,g^b=def,g^c=dg,h^a=hik,h^b=deh,h^c=hi,i^a=j,i^b=fgi,i^c=hj,j^a=i,j^b=gj,j^c=ik,k^a=ijk,k^b=dfk,k^c=hij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219852
maximal quotient of	#72272633

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