#72230518

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₄×M₁₆
Sylow	C₅² ⋊ Dic₃×M₁₆
Sylow	C₃ ⋊ #13103905

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×D₅⋊ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅²	C₂×C₄²
Fitting	C₂×C₃×C₄×C₅²	C₂×C₄
prime radical	C₂²×C₃×C₅⋊D₅	C₂×C₄
cosocle	C₃×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₅⋊ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅²	C₄×M₁₆
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

784 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 139 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:

2^6,3,5^2|a^2=e,c^2=d,e^2=f,b^a=bf,g^c=g^2,h^a=h^4,h^c=h^3,h^d=h^4,i^a=h^2i,i^c=i^3,i^d=i^4

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13103905, #72259593, #72259668, …
maximal normal subgroup of	D₃×#13103905, #72259593, #72259668, …

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