#72215875

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

Aliases

No aliases for this group.

Splittings

Fitting

#15947 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	#85645	C₂²
Frattini	C₂⁵	C₂×S₄
Fitting	#15947	D₃
socle	C₂⁴	GL(2,ℤ/4)
hypocenter	C₂²⋊(C₂²⋊A₄)	D₄
LCS	G ▷ #85645 ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁

1152 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 1152 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:

2^3,3,2^6|b^a=bc,d^b=d^2,e^a=eg,e^b=f,e^c=eij,e^d=f,f^a=fh,f^b=e,f^c=fi,f^d=ef,g^b=hi,g^d=h,h^b=gij,h^d=gh,i^b=ij,i^d=j,j^d=ij

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215886, C₂×#72215875, C₃×#72215875
maximal normal subgroup of	#72215886, C₂×#72215875, C₃×#72215875
maximal quotient of	#72216213, #72218023, C₂×#72215875, …

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