#72244258

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₉

Aliases

No aliases for this group.

Splittings

Fitting	C₉²×#97 ⋊ C₂
Sylow	C₉² ⋊ #442

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₉⋊D₉)
commutator	C₄×C₉²	C₂²×C₄
Frattini	C₂×C₃²×C₄	C₂²×C₃⋊D₃
Fitting	C₉²×#97	C₂
socle	C₂²×C₃²	C₂×C₄⋊(C₃⋊D₃)
prime radical	C₂×#13046732	C₂
cosocle	C₂×C₄×C₉²	C₂³
hypercenter	#97	C₉⋊D₉
hypocenter	C₉²	#442
LCS	G ▷ C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

1518 conjugacy classes of subgroups; only the 387 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 387 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) = 2592

Polycyclic presentation:

2^6,3^4|a^2=d,b^2=d,c^2=e,e^2=f,g^3=i,h^3=j,c^a=ce,c^b=cf,e^a=ef,g^a=g^2i^2,h^a=h^2j^2,i^a=i^2,j^a=j^2

Conjugacy classes & character table

822 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#442×C₉⋊D₉
maximal normal subgroup of	#442×C₉⋊D₉

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);