#72253802

order	3840
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₂⁶×C₅ ⋊ C₃×C₄
Hall	C₂²×#65800 ⋊ C₃
Sylow	C₅ ⋊ #11709759

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₂⁶×C₅	C₃×C₄
Frattini	C₂²	AGL(1,𝔽₅)×C₂²⋊A₄
socle	C₂⁴×C₅	C₄×A₄
prime radical	D₅×C₂²⋊(C₂²⋊A₄)	C₂
cosocle	C₂⁶×D₅	C₂×C₃
LCS	G ▷ C₂⁶×C₅
UCS	C₁

11298 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 11298 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 120

Polycyclic presentation:

2,3,2^7,5|a^2=c,d^b=e,e^b=de,f^a=fhi,f^b=g,g^a=gh,g^b=fg,h^b=i,i^b=hi,j^a=j^3,j^c=j^4

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72253825

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