#72215851

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#347042 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	C₂²×#120	C₂²×C₃
Frattini	C₂⁵	C₂²×A₄
Fitting	#347042	C₃
socle	C₂²	AΣL(1,𝔽₆₄)
hypocenter	C₂⁶	C₃×D₄
LCS	G ▷ C₂²×#120 ▷ C₂⁶
UCS	C₁

1396 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 22 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:

2^2,3,2^7|b^a=bd,e^a=eg,e^b=ej,e^c=ef,e^d=ej,f^a=fh,f^b=fi,f^c=e,f^d=fi,g^b=gj,g^c=gh,h^b=hi,h^c=g,i^c=j,j^c=ij

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215854, #72215886, C₂×#72215851, …
maximal normal subgroup of	#72215886, C₂×#72215851, C₃×#72215851
maximal quotient of	#72217282, C₂×#72215851, #72247716, …

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