#72247393

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₅×3₋¹⁺² ⋊ C₂²⋊C₄
Sylow	3₋¹⁺² ⋊ C₂²⋊AGL(1,𝔽₅)
Sylow	C₅ ⋊ (C₂×AGL(1,ℤ/9))⋊C₄

Subgroups

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

220 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₄⋊AGL(1,ℤ/9), AGL(1,𝔽₅)×#611⋊C₃, #72247452, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₄⋊AGL(1,ℤ/9), AGL(1,𝔽₅)×#611⋊C₃, #72247452, …
maximal quotient of	#72254709, #72254712, #72254726

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