#72222994

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×H(3,𝔽₃) ⋊ C₂²×D₃
derived	(C₃×A₄)⋊H(3,𝔽₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	(C₃×A₄)⋊H(3,𝔽₃)	C₂³
Frattini	C₃²	D₃²×S₄
Fitting	C₂²×C₃×H(3,𝔽₃)	C₂²×D₃
socle	C₂²×C₃²	D₃³
LCS	G ▷ (C₃×A₄)⋊H(3,𝔽₃)
UCS	C₁

2402 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 2402 conjugacy classes of subgroups — too many to draw. Only its 60 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:

2^3,3^5,2^2|d^b=d^2,d^c=d^2,e^a=e^2,e^b=e^2,e^d=eg,f^a=e^2f,f^b=f^2,f^d=fgh^2,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=j,i^b=j,i^e=ij,i^f=j,j^a=i,j^b=i,j^e=i,j^f=ij

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226353
maximal quotient of	#72226353

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