#72209702

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁷×C₄⁴

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#72209699 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#72209699	C₃
Frattini, socle	C₂⁴	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ #72209699
UCS	C₁

40768 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 40768 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^10|b^2=i,c^2=h,b^a=ci,c^a=bc,d^a=e,d^b=dj,d^c=djk,e^a=de,e^b=ejk,e^c=ek,f^a=g,f^b=fhi,f^c=fh,g^a=fg,g^b=gh,g^c=gi,h^a=hi,i^a=h,j^a=k,k^a=jk

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209703
maximal quotient of	#72210033, #72210042, #72219176, …

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