#72209781

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	#87588
commutator, prime radical, cosocle, hypocenter	C₂²⋊(#543⋊C₃)	C₂
Fitting	C₂²×#543	D₃
socle	C₂⁴	#1020
LCS	G ▷ C₂²⋊(#543⋊C₃)
UCS	C₁ ◁ C₂²

1476 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 29 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|a^2=ij,e^2=j,f^2=j,b^a=b^2,c^a=cd,c^b=d,d^b=cd,e^a=fj,e^b=f,f^a=ej,f^b=efj,f^e=fj,g^a=cdgh,g^b=h,g^e=gi,h^a=dh,h^b=gh,h^e=hi,h^f=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209858, #72209877, #72227788, …
maximal normal subgroup of	#72209858, #72209877, #72227788, …
maximal quotient of	#72209783, #72227836

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