#72227331

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#8097815 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂²×#560⋊C₃
commutator, cosocle, hypocenter	#95	C₂⁴×C₃
Frattini, socle	C₂³	C₂⁴×A₄
Fitting	#8097815	C₃
hypercenter	Q₈	#560⋊C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂ ◁ Q₈

2293 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2293 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^4,3,2^5|c^2=j,d^2=j,f^2=h,g^2=i,d^c=dj,e^d=ehj,f^a=fij,f^b=fhj,f^e=ghj,g^a=ghi,g^b=gij,g^e=fgj,g^f=gj,h^e=i,i^e=hij

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228147
maximal normal subgroup of	#72228147
maximal quotient of	#72227333

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