#72219733

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²⋊H(3,𝔽₃) ⋊ C₂³
Sylow	C₃²⋊H(3,𝔽₃) ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×((((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×C₃²⋊H(3,𝔽₃)	C₂⁴
Frattini, socle	C₂×C₃²	C₂×D₃³
Fitting	C₂²×C₃²⋊H(3,𝔽₃)	C₂³
hypercenter	C₂²	((((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃)⋊C₂
hypocenter	C₃²⋊H(3,𝔽₃)	C₂²×D₄
LCS	G ▷ C₂×C₃²⋊H(3,𝔽₃) ▷ C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂²

4805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3^5|b^a=be,f^c=f^2,g^a=g^2,g^c=gi^2,g^d=g^2,g^f=gi,h^a=gh,h^c=hj^2,h^d=h^2,h^f=hj,i^a=i^2,i^c=i^2,i^d=i^2,j^a=ij,j^c=j^2,j^d=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72219734

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