#72260706

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#158342 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#15480⋊C₃
commutator, cosocle, hypocenter	C₂²×#95	C₂²×C₃
Frattini, socle	C₂⁵	C₂²×A₄
Fitting	#158342	C₃
prime radical	#7555⋊C₃	C₂
LCS	G ▷ C₂²×#95
UCS	C₁ ◁ C₂

1462 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 48

Polycyclic presentation:

2^2,3,2^7|a^2=j,d^2=h,e^2=i,c^b=cij,d^a=dfhj,d^b=dfgij,d^c=ehj,e^a=eg,e^b=ef,e^c=dej,e^d=ej,f^c=gij,g^c=fghj,h^c=i,i^c=hij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260766, C₂×#72260706, #72261431, …
maximal normal subgroup of	C₂×#72260706, #72261431, #72261766
maximal quotient of	C₂×#72260706, #72299986

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