#72218551

order	3840
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₂⁸ ⋊ C₃×C₅
Hall	C₂⁴⋊(C₂⁴⋊C₅) ⋊ C₃
Hall	C₂⁴×C₂²⋊A₄ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁸	C₃×C₅
LCS	G ▷ C₂⁸
UCS	C₁

27993 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 27993 conjugacy classes of subgroups — too many to draw. Only its 8 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:

3,5,2^8|c^a=cd,c^b=f,d^a=c,d^b=ce,e^a=cdf,e^b=d,f^a=cef,f^b=ef,g^a=cg,g^b=i,h^a=eh,h^b=ghj,i^a=fi,i^b=hi,j^a=cdej,j^b=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218553
maximal normal subgroup of	#72218553
maximal quotient of	#72272552

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