#72269037

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₉×#1790 ⋊ C₂
Sylow	C₉ ⋊ #32898

Subgroups

	subgroup	G/H
center	C₂²	D₉×#120
commutator, cosocle	C₂⁴×C₉	C₂⁴
Frattini	C₂⁴×C₃	C₂³×D₃
Fitting	C₉×#1790	C₂
socle	C₂²×C₃	D₃×#120
hypercenter	#1790	D₉
hypocenter	C₉	#32898
LCS	G ▷ C₂⁴×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #1790

6540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Polycyclic presentation:

2^8,3^2|b^2=e,d^2=h,i^3=j,b^a=bf,c^a=cf,c^b=ce,d^a=dh,d^b=dg,d^c=dgh,f^d=fh,g^a=gh,i^c=i^2j^2,j^c=j^2

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

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