#72220519

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²×C₃⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃³ ⋊ C₃⋊D₃
derived	C₃²×A₄² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×A₄²	C₂×C₃
Frattini	C₃²	C₃×A₄⋊S₄
Fitting	C₂⁴×C₃³	C₃⋊D₃
socle	C₂⁴×C₃²	C₃×C₃⋊D₃
LCS	G ▷ C₃²×A₄²
UCS	C₁

1758 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1758 conjugacy classes of subgroups — too many to draw. Only its 62 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:

2,3^5,2^4|c^a=c^2,c^b=ce,d^a=d^2,d^b=def^2,e^a=e^2,f^a=f^2,g^c=h,g^d=gh,h^a=gh,h^c=gh,h^d=g,i^a=gi,i^c=gi,i^d=j,j^a=hij,j^c=ghj,j^d=ij

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222920, #72222968, #72222976
maximal normal subgroup of	#72222920, #72222968, #72222976

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