#72238907

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#8093169 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11710658
commutator, Fitting, cosocle, hypocenter	#8093169	C₃
Frattini, socle	C₂³	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ #8093169
UCS	C₁ ◁ C₂

1613 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1613 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

3,2^9|b^2=h,c^2=hij,d^2=ij,e^2=i,f^2=i,g^2=ij,b^a=ci,c^a=bchi,c^b=chi,d^a=eij,d^b=dij,d^c=dhij,e^a=de,e^b=ehij,e^c=ei,f^a=fg,f^b=fi,f^c=fij,g^a=fij,g^b=gj,g^c=gi,h^a=hij,i^a=j,j^a=ij

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238908, #72239921
maximal normal subgroup of	#72238908, #72239921

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