#72209808

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	#15467 ⋊ D₃
derived	#15467⋊C₃ ⋊ C₂

Subgroups

	subgroup	G/H
center	C₂	#11714279
commutator, cosocle, hypocenter	#15467⋊C₃	C₂
Frattini	C₂⁴	#1020
Fitting	#15467	D₃
socle	C₂³	#5934
hypercenter	C₂²	#85559
LCS	G ▷ #15467⋊C₃
UCS	C₁ ◁ C₂ ◁ C₂²

735 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 12 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:

2,3,2^8|c^2=g,d^2=h,e^2=ghij,f^2=hij,b^a=b^2,c^a=dgh,c^b=dg,d^a=cgh,d^b=cd,e^a=cefghij,e^b=fh,e^c=ehi,e^d=ehj,f^a=cdfh,f^b=efgij,f^c=fghij,f^d=fi,f^e=fgij,g^a=h,g^b=h,h^a=g,h^b=gh,i^a=ghij,i^b=ghi

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209891
maximal normal subgroup of	#72209891
maximal quotient of	#72209810, #72209818, #72227659, …

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