#72249279

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#61519 ⋊ C₂
Sylow	C₃ ⋊ #10576928

Subgroups

	subgroup	G/H
center	C₂²	C₂³×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁶
Frattini	C₂×C₄	C₂⁵×D₃
Fitting	C₃×#61519	C₂
socle	C₂²×C₃	C₂⁴×D₄
hypercenter	#61519	D₃
hypocenter	C₃	#10576928
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #529 ◁ #61519

13748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|c^2=h,f^2=hi,g^2=i,b^a=bi,d^a=dh,f^c=fg,f^e=fgi,g^c=gi,g^e=gi,g^f=gi,j^b=j^2,j^e=j^2,j^f=j^2

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#10576928
maximal normal subgroup of	D₃×#10576928

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