#72224447

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72221195 ⋊ C₃
derived	#72219166 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72209713
commutator, cosocle, hypocenter	#72219166	C₂×C₃
Frattini	C₂⁶	C₂×C₂²⋊A₄
Fitting	#72221195	C₃
socle	C₂⁴	#85645
LCS	G ▷ #72219166
UCS	C₁ ◁ C₂²

At least 22384 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has at least 22384 conjugacy classes of subgroups — too many to draw. Only its 46 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:

2,3,2^10|e^2=i,f^2=k,c^b=d,d^b=cd,e^a=eh,e^b=ef,e^c=ek,e^d=ekl,f^a=fg,f^b=eijkl,f^c=fij,f^d=fikl,f^e=fijl,g^b=h,g^e=gil,h^b=ghil,h^f=hil,i^b=jkl,k^b=i,l^b=ijk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224460
maximal quotient of	#72267024, #72267061, #72267346, …

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", "l");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.11);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.5*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.12);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.12);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.10*F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.9);;
gap> SetConjugate(coll, 12, 2, F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);