#72259090

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁵×C₄ ⋊ D₃²

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×S₄²
commutator, cosocle	C₂×A₄²	C₂⁴
Fitting	C₂⁵×C₄	D₃²
socle	C₂⁵	C₂²×D₃²
hypercenter	C₂×C₄	S₄²
hypocenter	A₄²	H(5,𝔽₂)
LCS	G ▷ C₂×A₄² ▷ A₄²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

11675 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 11675 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

2^5,3^2,2^4|b^2=e,b^a=be,c^a=ce,d^b=de,f^a=f^2,f^d=f^2,g^a=f^2g,g^d=g^2,h^f=hi,h^g=i,i^a=hi,i^d=hi,i^f=h,i^g=hi,j^a=hj,j^d=hik,j^f=hj,j^g=k,k^d=ij,k^f=ik,k^g=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

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