#72259883

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

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₂×#104
Sylow	C₅² ⋊ C₂×#837
Sylow	C₃ ⋊ #13100574

Subgroups

	subgroup	G/H
center	C₄	#11925420
commutator	C₃×C₄×C₅²	C₂²×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅²
Fitting, cosocle	C₂×C₃×C₄×C₅²	C₂³
socle	C₂×C₃×C₅²	C₂×C₂²⋊C₄
prime radical	C₃×#11717471	C₂
hypocenter	C₃×C₅²	C₂×#104
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

896 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 126 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=f,b^2=d,e^2=f,b^a=be,d^a=df,e^a=ef,g^b=g^2,h^a=h^4,h^c=h^4,i^a=h^3i,i^c=i^4

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259938, D₃×#13100574
maximal normal subgroup of	#72259938, D₃×#13100574

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