#72214726

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

solvable

Aliases

No aliases for this group.

Splittings

Fitting

#11337 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	#73005
commutator, hypocenter	#338⋊C₃	C₂×C₄
Frattini	C₂³×C₄	C₂×S₄
Fitting	#11337	D₃
socle	C₂	#11711091
prime radical	#11711358	C₂
cosocle	C₂×#338⋊C₃	C₂²
hypercenter	C₂×C₄	#857⋊C₂
LCS	G ▷ #338⋊C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

346 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 25 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:

2^3,3,2^6|b^2=c,c^2=j,e^2=g,f^2=h,g^2=j,h^2=j,i^2=j,b^a=bhij,d^a=d^2,d^b=dhi,e^a=efgij,e^b=egij,e^d=fgij,f^a=fij,f^b=fhi,f^d=efi,f^e=fi,g^a=ghij,g^d=hj,g^f=gj,h^a=hj,h^d=ghij,h^e=hj,i^a=ij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218424
maximal normal subgroup of	#72218424
maximal quotient of	#72214741, #72214754, #72215445

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