#72258646

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72258564 ⋊ C₃
derived	C₂²×#62817 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center	C₂	#72258302
commutator, cosocle, hypocenter	C₂²×#62817	C₂×C₃
Frattini	C₂⁶	C₂×C₂²⋊A₄
Fitting	#72258564	C₃
socle	C₂³	#35346⋊C₃
hypercenter	C₂²	#72215614
LCS	G ▷ C₂²×#62817
UCS	C₁ ◁ C₂ ◁ C₂²

At least 9669 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has at least 9669 conjugacy classes of subgroups — too many to draw. Only its 53 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|c^a=cg,c^b=d,d^a=dh,d^b=cd,e^a=ei,e^b=f,e^c=ek,e^d=ek,f^a=fj,f^b=ef,f^d=fk,g^b=h,g^e=gl,h^b=gh,h^e=hl,h^f=hl,i^b=j,j^b=ij,k^a=kl

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258797
maximal normal subgroup of	#72258797

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