#72267166

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72266018 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#85645
commutator, cosocle	C₂³×H(5,𝔽₂)	C₂³×C₃
Frattini, socle	C₂⁴	C₂³×C₂²⋊A₄
Fitting	#72266018	C₃
hypercenter	C₂×D₄	#85645
hypocenter	C₂²×H(5,𝔽₂)	C₂×C₃×D₄
LCS	G ▷ C₂³×H(5,𝔽₂) ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂×D₄

At least 30437 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^8|b^2=l,b^a=bl,c^b=ce,f^d=fg,g^d=f,h^c=hj,h^d=hi,h^g=hl,i^c=ik,i^d=h,i^f=il,j^d=jk,k^d=j

Conjugacy classes & character table

160 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, 2, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.8);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);