#72213802

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#6342426 ⋊ C₃²
derived	C₄⁴ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄⁴	C₂×C₃²
Frattini, socle	C₂⁴	C₂×A₄²
Fitting	#6342426	C₃²
LCS	G ▷ C₄⁴
UCS	C₁

1820 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2,3^2,2^8|d^2=h,e^2=i,f^2=j,g^2=k,d^a=dh,d^b=de,d^c=ehi,e^a=ei,e^b=di,e^c=deh,f^a=fj,f^b=gj,f^c=efhi,g^a=gk,g^b=fg,g^c=dgi,h^b=hi,h^c=i,i^b=h,i^c=hi,j^b=k,j^c=ij,k^b=jk,k^c=hk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214174, #72214317, #72219739, …
maximal normal subgroup of	#72214174, #72214317, #72219739, …
maximal quotient of	#72222676, C₂×#72213802

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