#72259392

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ #417
Sylow	C₅² ⋊ #5232
Sylow	C₃ ⋊ #13103931

Subgroups

	subgroup	G/H
center	C₂	C₂×#11925350
commutator	C₃×C₄×C₅²	C₂²×C₄
Frattini	C₄	C₂×D₅⋊ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄×C₅²	C₂×C₄
socle	C₂×C₃×C₅²	C₂×C₂²⋊C₄
prime radical	C₃×#11717471	C₂
cosocle	C₃×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₅⋊ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅²	#417
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

832 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 119 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=d,b^2=de,d^2=f,e^2=f,b^a=be,c^a=cf,d^b=df,e^b=ef,g^a=g^2,g^b=g^2,h^a=h^2,h^b=h^3,h^d=h^4,i^a=h^3i^3,i^b=i^3,i^d=i^4

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259424, #72259593, D₃×#13103931
maximal normal subgroup of	#72259424, #72259593, D₃×#13103931

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