#72231503

order	7200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅² ⋊ C₄×D₄
Sylow	C₅² ⋊ C₄×#638
Sylow	C₃² ⋊ #11717375

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊D₃×D₅⋊AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₃²×C₅²	C₂²×C₄
prime radical	#72229358	C₂
cosocle	C₂×C₃²×C₅⋊D₅	C₂³
hypocenter	C₃²×C₅²	C₄×D₄
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂

1452 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1452 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#11717375, C₂×#72231503
maximal normal subgroup of	C₃⋊D₃×#11717375, C₂×#72231503
maximal quotient of	C₂×#72231503

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, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, 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.8^3*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);