#72216927

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72216427 ⋊ C₃
derived	#62817 ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, socle	C₂	#72216921
commutator, hypocenter	#62817	C₃×C₄
Frattini	C₂⁷	C₂×A₄
Fitting	#72216427	C₃
prime radical	#72209717	C₂
cosocle	#6499867	C₂×C₃
hypercenter	C₂²	#8541⋊C₃
LCS	G ▷ #62817
UCS	C₁ ◁ C₂ ◁ C₂²

2546 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 2546 conjugacy classes of subgroups — too many to draw. Only its 17 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:

2,3,2^9|a^2=c,d^2=k,e^2=k,d^a=dg,d^b=e,d^c=di,e^a=ef,e^b=dek,e^c=eh,e^d=ek,f^a=fh,f^b=fgj,f^d=fj,g^a=gi,g^b=f,g^e=gj,h^b=hik,h^d=hk,i^b=h,i^e=ik,j^a=jk

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218442, #72218459, C₂×#72216927
maximal normal subgroup of	#72217355, #72218442, #72218459, …
maximal quotient of	#72241400, C₂×#72216927

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