#72222847

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#1079013 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#560⋊C₃
commutator, cosocle	C₂×#95	C₂³×C₃
Frattini, socle	C₂⁴	C₂³×A₄
Fitting	#1079013	C₃
hypercenter	C₂×C₄	#560⋊C₃
hypocenter	#95	C₃×#41
LCS	G ▷ C₂×#95 ▷ #95
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1071 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 62 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:

2^3,3,2^6|a^2=e,c^2=ej,f^2=h,g^2=i,b^a=bj,c^a=ce,c^b=ce,d^a=dhi,f^a=fhi,f^b=fhj,f^d=fg,g^a=ghj,g^b=gij,g^d=fij,g^f=gj,h^d=hij,i^d=h

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227334
maximal normal subgroup of	#72227334
maximal quotient of	#72222855

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