#72277479

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Hall	#66285 ⋊ C₃²
Sylow	C₅ ⋊ (#41⋊C₃)⋊A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×A₄²
commutator, Fitting, cosocle, hypocenter	C₅×H(5,𝔽₂)	C₂×C₃²
socle	C₂×C₅	C₂×A₄²
LCS	G ▷ C₅×H(5,𝔽₂)
UCS	C₁ ◁ C₂

276 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 240

Polycyclic presentation:

2,3^2,2^4,5,2|a^2=i,d^2=i,e^2=i,d^b=e,d^c=e,e^b=dei,e^c=dei,e^d=ei,f^b=g,f^c=def,f^e=fi,g^b=fg,g^c=dgi,g^d=gi,h^a=h^4

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×(#41⋊C₃)⋊A₄
maximal normal subgroup of	D₅×(#41⋊C₃)⋊A₄

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