#72247462

order	9720
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×#6891 ⋊ C₂×C₄
derived	C₅×C₃²⋊C₉ ⋊ C₂×C₃×C₄
Sylow	#6891 ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #92259⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×C₃²⋊C₉	C₂×C₃×C₄
Frattini	C₃²	C₃×D₃×ΓL(1,𝔽₁₆)
Fitting	C₅×#6891	C₂×C₄
socle	C₃²×C₅	C₃×D₃×Dic₃
prime radical	C₃²×D₅×C₃⋊D₃	D₃
cosocle	D₅×C₃²⋊C₉	C₂²×C₃
LCS	G ▷ C₅×C₃²⋊C₉
UCS	C₁

600 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 90

Polycyclic presentation:

2^2,3,2,3^2,5,3^2|b^2=d,f^3=i,e^a=e^2,e^b=e^2,f^b=f^2i^2,f^c=fi,f^e=fh,g^b=g^3,g^d=g^4,h^a=h^2,i^b=i^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃
maximal normal subgroup of	AGL(1,𝔽₅)×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃

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