#72247452

order	4320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅×3₋¹⁺² ⋊ C₂×C₄
Hall	C₅×3₋¹⁺² ⋊ C₄×D₄
Sylow	3₋¹⁺² ⋊ D₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #63455⋊C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×AGL(1,𝔽₅)×AGL(1,ℤ/9)
commutator	C₂×C₅×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃	C₂×C₃×D₃×AGL(1,𝔽₅)
Fitting	C₄×C₅×3₋¹⁺²	C₂×C₄
socle	C₂×C₃×C₅	C₂×C₃×C₄×D₃
prime radical	D₅×C₄⋊AGL(1,ℤ/9)	C₂
cosocle	C₂×C₉×D₅	C₂³×C₃
hypercenter	C₄	AGL(1,𝔽₅)×AGL(1,ℤ/9)
hypocenter	C₅×C₉	C₃×C₄×D₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄

590 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 180

Polycyclic presentation:

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

Conjugacy classes & character table

131 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,𝔽₅)×AGL(1,ℤ/9), #72269526
maximal normal subgroup of	D₄×AGL(1,𝔽₅)×AGL(1,ℤ/9)
maximal quotient of	#72269526

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