#72209907

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#7622914 ⋊ C₃
derived	#63040 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, socle	C₂	#11710337
commutator, cosocle, hypocenter	#63040	C₂×C₃
Frattini	C₂×Q₈	C₂×C₂²⋊A₄
Fitting	#7622914	C₃
hypercenter	C₂²	#85645
LCS	G ▷ #63040
UCS	C₁ ◁ C₂ ◁ C₂²

1500 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 1500 conjugacy classes of subgroups — too many to draw. Only its 24 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|a^2=j,c^2=i,d^2=i,g^2=j,h^2=j,c^a=cg,c^b=cd,d^a=dh,d^b=c,d^c=di,e^b=ef,e^c=ei,f^b=e,f^c=fi,f^d=fi,g^a=gj,g^b=ghi,g^d=gi,g^e=gj,g^f=gj,h^a=hj,h^b=g,h^c=hi,h^f=hj,h^g=hj,i^a=ij

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209911, #72259006
maximal normal subgroup of	#72209911
maximal quotient of	#72227584, #72227831, #72228023, …

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