#72218273

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

solvable

Aliases

No aliases for this group.

Splittings

Fitting

C₄×#392 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#857⋊C₂
commutator, hypocenter	#95⋊C₃	C₂²×C₄
Frattini, socle	C₂⁴	C₂²×S₄
Fitting	C₄×#392	D₃
prime radical	C₂×#72996	C₂
cosocle	C₂×#95⋊C₃	C₂³
hypercenter	C₂×C₄	#857⋊C₂
LCS	G ▷ #95⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1192 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 52 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:

2^4,3,2^5|a^2=j,b^2=j,c^2=d,f^2=h,g^2=i,c^a=ci,e^a=e^2,e^c=eij,f^a=fghj,f^b=fhj,f^e=ghj,g^a=gj,g^b=gij,g^e=fgj,g^f=gj,h^a=hij,h^e=i,i^e=hij

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72218286, #72218323

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