#72261456

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#10586400 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#11715850
commutator, cosocle	C₂×#11710852	C₂²
Frattini	C₂³	C₂×#87588
Fitting	C₂×#10586400	D₃
hypercenter	C₂²×C₄	#87588
hypocenter	#11710852	D₄
LCS	G ▷ C₂×#11710852 ▷ #11710852
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Only the 67 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 67 normal subgroups are shown; non-normal subgroups are omitted.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^8|a^2=c,b^2=l,g^2=l,h^2=l,b^a=bc,d^a=d^2,e^a=ef,e^d=f,f^d=ef,g^a=hl,g^d=h,h^a=gl,h^d=ghl,h^g=hl,i^d=j,i^e=ik,i^f=il,j^a=ij,j^d=ij,j^e=jkl,j^f=jk,k^a=kl

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

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