#72245409

order	3072
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #72224644

Subgroups

	subgroup	G/H
center	C₄	#10681804
commutator	C₃×C₁₆	C₄×C₁₆
Frattini	#7330	C₂×D₃
Fitting	C₃×#94509	C₂
socle	C₂×C₃	#93668
prime radical	C₂²×C₃	#7333
cosocle	C₃×#7330	C₂²
hypercenter	#94509	D₃
hypocenter	C₃	#72224644
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₄×C₁₆ ◁ #94509

150 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 192

Polycyclic presentation:

2^10,3|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=chj,d^b=dh,d^c=dij,e^a=ei,e^d=ej,f^b=fi,f^c=fj,g^a=gj,h^b=hj,k^a=k^2,k^b=k^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72224644, #72269698, #72269705
maximal normal subgroup of	D₃×#72224644, #72269698, #72269705

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);