#72267016

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#154039 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	C₂²⋊(C₄²⋊C₃)
commutator, Fitting, cosocle, hypocenter	#154039	C₃
Frattini, socle	C₂⁵	C₂²⋊A₄
LCS	G ▷ #154039
UCS	C₁ ◁ C₂³

1328 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 192

Polycyclic presentation:

3,2^9|b^2=f,c^2=fh,d^2=j,e^2=j,b^a=cg,c^a=bcfg,c^b=cfg,d^a=de,d^b=dfhi,e^a=d,e^c=efhi,e^d=ej,f^a=fh,g^a=gh,h^a=g,i^a=gi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267018, #72267022, #72267047, …
maximal normal subgroup of	#72267022
maximal quotient of	#72267722

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