#72267652

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#4131221 ⋊ C₃
derived	#17309 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#3549⋊C₃
commutator, cosocle, hypocenter	#17309	C₂×C₃
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	#4131221	C₃
LCS	G ▷ #17309
UCS	C₁ ◁ C₂²

799 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 96

Polycyclic presentation:

2,3,2^8|c^2=g,d^2=ghi,e^2=j,f^2=j,b^a=bh,c^a=ch,c^b=cd,d^a=di,d^b=ci,d^c=dgh,e^a=ei,e^b=f,e^c=eh,e^d=ehij,f^a=fh,f^b=efj,f^c=fij,f^d=fhj,f^e=fj,g^b=gi,h^b=hi,i^b=h

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267664
maximal normal subgroup of	#72267664
maximal quotient of	#72267657

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