#72267618

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

Q₈×#2840 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	C₂²⋊#5464
commutator, Fitting, cosocle, hypocenter	Q₈×#2840	C₃
Frattini, socle	C₂⁴	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ Q₈×#2840
UCS	C₁ ◁ C₂²

6325 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 6325 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 192

Polycyclic presentation:

3,2^10|d^2=j,e^2=j,f^2=k,g^2=k,b^a=bc,c^a=b,d^a=de,d^b=dh,d^c=di,e^a=d,e^b=ei,e^c=ehi,e^d=ej,f^a=fg,g^a=f,g^f=gk,h^a=i,i^a=hi

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267621

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