#72245378

order	1536
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #94509

Subgroups

	subgroup	G/H
center	C₈	#4455
commutator	C₃×C₈	C₄×C₁₆
Frattini	#1397	C₂×D₃
Fitting	C₃×#7330	C₂
socle	C₂×C₃	#7332
prime radical	C₂²×C₃	#1398
cosocle	C₃×#1397	C₂²
hypercenter	#7330	D₃
hypocenter	C₃	#94509
LCS	G ▷ C₃×C₈ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ #1397 ◁ #7330

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=h,h^2=i,b^a=bfhi,c^b=chi,d^a=dhi,d^c=di,e^b=ei,f^a=fi,j^a=j^2

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#94509, #72245402, #72245406, …
maximal normal subgroup of	D₃×#94509, #72245402, #72245406, …

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