#72209796

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#475912 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#5464
commutator, Fitting, cosocle, hypocenter	#475912	C₃
Frattini	C₂³×C₄	C₂²⋊A₄
socle	C₂²	#2679⋊C₃
LCS	G ▷ #475912
UCS	C₁ ◁ C₂×C₄

401 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 21 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|b^2=f,c^2=f,d^2=f,e^2=f,h^2=j,b^a=c,c^a=bcf,c^b=cf,d^a=e,d^b=dh,d^c=dg,e^a=def,e^b=eghij,e^c=eghj,e^d=ef,g^a=ij,g^b=gj,g^e=gj,h^a=ghj,h^b=hj,h^c=hj,h^d=hj,i^a=gij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227661
maximal subgroup of	#72209802, #72209826, #72227661, …
maximal normal subgroup of	#72209802, #72209826, #72227661, …
maximal quotient of	#72227641, #72268642

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