#72235730

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#72235525 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#13829⋊C₃
commutator, Fitting, cosocle, hypocenter	#72235525	C₃
Frattini, socle	C₂⁶	C₂²⋊A₄
LCS	G ▷ #72235525
UCS	C₁ ◁ C₂²

5130 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 5130 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 48

Polycyclic presentation:

3,2^10|b^2=f,c^2=g,d^2=fgk,e^2=fj,b^a=bc,c^a=bgh,c^b=ch,d^a=de,d^b=di,d^c=dhij,e^a=dfhijk,e^b=eij,e^c=ehijk,e^d=ehik,f^a=fgh,g^a=f,i^a=jk,j^a=hk,k^a=ij

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72235731

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