#72230452

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ #417
Sylow	C₃² ⋊ #65793
Sylow	C₅ ⋊ #10590612

Subgroups

	subgroup	G/H
center	C₂	#13088405
commutator, Fitting	C₃²×C₄×C₅	C₂²×C₄
Frattini, hypercenter	C₄	D₃²×AGL(1,𝔽₅)
socle	C₂×C₃²×C₅	C₂×C₂²⋊C₄
prime radical	D₅×#3796	C₂
cosocle	C₃²×C₄×D₅	C₂³
hypocenter	C₃²×C₅	#417
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

740 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 98 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:

2^6,3^2,5|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^c=g^2,h^a=g^2h,h^c=h^2,i^a=i^3,i^b=i^3,i^d=i^4

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230455, AGL(1,𝔽₅)×#63937, #72230460
maximal normal subgroup of	#72230455, AGL(1,𝔽₅)×#63937, #72230460

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, 3, 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, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, 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);