#72258669

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁵×D₄ ⋊ D₃²

Subgroups

	subgroup	G/H
center	C₂	C₂×#72215293
commutator, cosocle	C₄×A₄²	C₂⁴
Frattini	C₄	C₂²×S₄²
Fitting	C₂⁵×D₄	D₃²
socle	C₂⁵	C₂×#3811
hypercenter	C₂×D₄	S₄²
hypocenter	A₄²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×A₄² ▷ C₂×A₄² ▷ A₄²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

At least 9121 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3^2,2^4|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^d=g^2,h^a=g^2h,h^d=h^2,i^a=j,i^d=j,i^g=ij,i^h=j,j^a=i,j^d=i,j^g=i,j^h=ij,k^a=ijk,k^d=ijk,k^g=ijk,k^h=l,l^d=jkl,l^g=il,l^h=kl

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

holomorph of

A₄⋊C₈

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> SetConjugate(coll, 10, 7, F.9);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> SetConjugate(coll, 11, 1, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 7, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 8, F.12);;
gap> SetConjugate(coll, 12, 4, F.10*F.11*F.12);;
gap> SetConjugate(coll, 12, 7, F.9*F.12);;
gap> SetConjugate(coll, 12, 8, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);