#72247404

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×3₋¹⁺² ⋊ C₄
Hall	C₅×3₋¹⁺² ⋊ C₂²⋊C₄
Sylow	3₋¹⁺² ⋊ C₂²⋊AGL(1,𝔽₅)
Sylow	C₅ ⋊ #3676⋊C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₉	C₂×C₃×C₄
Frattini	C₂×C₃	C₂×C₃×ΓL(1,𝔽₁₆)
Fitting	C₂²×C₅×3₋¹⁺²	C₄
socle	C₂×C₃×C₅	C₂×C₃×Dic₃
prime radical	C₂²×C₃²×D₅	D₃
cosocle	C₂×C₉×D₅	C₂²×C₃
hypercenter	C₂²	3₋¹⁺²⋊AGL(1,𝔽₅)
hypocenter	C₅×C₉	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₂²

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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^a=f^2h^2,f^c=fh,g^a=g^3,g^d=g^4,h^a=h^2

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#611⋊C₃, D₄×3₋¹⁺²⋊AGL(1,𝔽₅), AGL(1,ℤ/9)×C₂²⋊AGL(1,𝔽₅), …
maximal normal subgroup of	AGL(1,𝔽₅)×#611⋊C₃, D₄×3₋¹⁺²⋊AGL(1,𝔽₅), AGL(1,ℤ/9)×C₂²⋊AGL(1,𝔽₅)
maximal quotient of	#72254706, #72254730

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, 1, 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, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);