#72268892

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₉×#1437 ⋊ C₂
Sylow	C₉ ⋊ #15258

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₉
commutator, cosocle	C₂⁴×C₉	C₂⁴
Frattini, socle	C₂⁴×C₃	C₂³×D₃
Fitting	C₉×#1437	C₂
hypercenter	#1437	D₉
hypocenter	C₉	#15258
LCS	G ▷ C₂⁴×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ #1437

7701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 576

Polycyclic presentation:

2^8,3^2|a^2=f,d^2=h,i^3=j,b^a=bf,c^a=ce,d^b=dh,d^c=dg,i^b=i^2j^2,j^b=j^2

Conjugacy classes & character table

282 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#15258
maximal normal subgroup of	D₉×#15258

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