#72230832

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

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₄×D₈
Sylow	C₅² ⋊ C₄×#213
Sylow	C₃ ⋊ #13104055

Subgroups

	subgroup	G/H
center	C₂	#72224586
commutator, Fitting	C₃×C₄×C₅²	C₂²×C₄
Frattini, hypercenter	C₄	D₃×D₅⋊AGL(1,𝔽₅)
socle	C₂×C₃×C₅²	C₄×D₄
prime radical	#72230828	C₂
cosocle	C₃×C₄×C₅⋊D₅	C₂³
hypocenter	C₃×C₅²	C₄×D₈
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄

724 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 84 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=be,e^b=ef,g^a=g^2,g^b=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

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13104055, #72230980, #72259424
maximal normal subgroup of	D₃×#13104055, #72230980, #72259424

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.5);;
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.8^4);;
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);