#72209656

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	#35390 ⋊ D₃
derived	#2273⋊C₃ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11715690
commutator, cosocle, hypocenter	#2273⋊C₃	C₂²
Frattini, socle	C₂³	C₂×#1020
Fitting	#35390	D₃
LCS	G ▷ #2273⋊C₃
UCS	C₁ ◁ C₂

2163 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 2163 conjugacy classes of subgroups — too many to draw. Only its 22 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:

2^2,3,2^7|f^2=h,g^2=i,c^b=c^2,d^b=e,d^c=e,e^b=d,e^c=de,f^a=fh,f^b=gh,f^c=gh,f^e=fj,g^a=gi,g^b=fi,g^c=fg,g^d=gj,h^b=i,h^c=i,i^b=h,i^c=hi

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209657, #72209792, #72255725, …
maximal normal subgroup of	#72209792, #72255999, #72341125
maximal quotient of	#72209856, #72209863

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