#72222151

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

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₅² ⋊ Dic₃
derived	C₅²×C₂²⋊A₄ ⋊ C₄
Sylow	C₅² ⋊ #5936

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₂²⋊A₄	C₄
Fitting, socle	C₂⁴×C₅²	Dic₃
prime radical, cosocle	C₅×D₅×C₂²⋊A₄	C₂
LCS	G ▷ C₅²×C₂²⋊A₄
UCS	C₁

634 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,5^2,2^4|a^2=b,c^a=c^2,d^a=d^3,d^b=d^4,e^a=d^2e^4,e^b=d^4e,f^a=fg,f^c=g,g^c=fg,h^c=i,i^a=hi,i^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#11757174, #72258493, #72258511
maximal normal subgroup of	D₅×#11757174, #72258493, #72258511

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