#72259907

order	2400
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ #95
Sylow	C₅² ⋊ C₃⋊#95
Sylow	C₃ ⋊ #11716776

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	#10597183
commutator	C₂×C₃×C₅²	C₄²
Fitting, socle	C₂²×C₃×C₅²	C₂×C₄
prime radical, cosocle	C₂²×C₃×C₅⋊D₅	C₂²
hypocenter	C₃×C₅²	#95
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂²

412 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259924
maximal normal subgroup of	#72259924
maximal quotient of	#72259915

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