#72210027

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(9,𝔽₂) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, Fitting, cosocle, hypocenter	H(9,𝔽₂)	C₃
LCS	G ▷ H(9,𝔽₂)
UCS	C₁ ◁ C₂

141405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^9|d^2=j,e^2=j,f^2=j,g^2=j,h^2=j,i^2=j,b^a=c,c^a=bc,d^a=e,d^b=dj,d^c=dj,e^a=dej,e^c=ej,e^d=ej,f^a=g,f^d=fj,g^a=fgj,g^d=gj,g^e=gj,g^f=gj,h^a=i,h^c=hj,i^a=hij,i^b=ij,i^h=ij

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₂×#72210031, #72256770, C₂×#72210040, …
maximal subgroup of	#72210031, #72210033, #72210040, …
maximal normal subgroup of	#72210031, #72210040, C₂×#72210027, …
maximal quotient of	#72241426, C₂×#72210027, #72258992, …

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