#72209758

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#7641846 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂³	C₂²⋊(C₂²⋊A₄)
commutator, Fitting, cosocle, hypocenter	#7641846	C₃
LCS	G ▷ #7641846
UCS	C₁ ◁ C₂³

4885 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 4885 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^9|d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=c,c^a=bc,d^a=e,d^b=dh,d^c=dhj,e^a=dehi,e^b=ej,e^c=eh,e^d=ehi,f^a=g,f^b=fj,f^c=fh,f^d=fij,g^a=fghi,g^b=ghj,g^c=gj,g^d=gij,g^e=gij,g^f=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227603, #72227751, #72227789, …
maximal subgroup of	#72209763, #72209783, #72227293, …
maximal normal subgroup of	#72209763, #72209783, #72227293, …
maximal quotient of	#72227587, #72267893, #72267935, …

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