#72343675

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#10586999 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×#1020
commutator, cosocle	C₂×#5950	C₂⁴
Frattini	C₂×C₄	C₂³×#1020
Fitting	C₂×#10586999	D₃
hypercenter	C₂×AGL(1,ℤ/8)	#1020
hypocenter	#997	C₂×#120
LCS	G ▷ C₂×#5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Only the 298 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^5|d^2=f,e^2=l,f^2=l,d^a=dl,d^b=de,d^c=defl,e^b=el,f^b=fl,g^c=g^2,g^d=g^2,h^c=hi,h^d=hi,h^g=i,i^g=hi,j^c=jk,j^d=jk,j^g=k,j^i=jl,k^g=jk,k^h=kl

Conjugacy classes & character table

158 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, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.12);;
gap> SetConjugate(coll, 11, 7, F.10*F.11);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);