#72209848

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#6499877 ⋊ C₃
derived	#62871 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, socle	C₂	#35346⋊C₃
commutator, cosocle, hypocenter	#62871	C₂×C₃
Frattini	C₂⁴	C₂×C₂²⋊A₄
Fitting	#6499877	C₃
hypercenter	C₂²	#85645
LCS	G ▷ #62871
UCS	C₁ ◁ C₂ ◁ C₂²

1413 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 1413 conjugacy classes of subgroups — too many to draw. Only its 26 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:

2,3,2^8|c^2=j,d^2=j,e^2=j,f^2=j,c^b=cd,d^b=c,d^c=dj,e^a=eg,e^b=ef,e^c=ej,e^d=ei,f^a=fh,f^b=e,f^c=fij,f^d=fj,f^e=fj,g^b=gh,g^d=gj,h^b=g,h^c=hj,i^a=ij

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209851
maximal normal subgroup of	#72209851
maximal quotient of	#72227299, #72227667, #72227704, …

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