#72341691

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72312874 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	C₄×#379⋊C₃
commutator	#585	C₃×C₄²
Frattini	C₂⁴×C₄	C₂²×A₄
Fitting	#72312874	C₃
socle	C₂	#72341331
prime radical	C₂²×#5950	C₂²
cosocle	C₂²×#585	C₂²×C₃
hypercenter	C₂×M₁₆	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₃×#95
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

10716 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 10716 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^2,3,2^8|a^2=d,b^2=f,e^2=k,b^a=bdek,d^b=dk,e^b=ek,f^a=fk,g^a=gi,g^b=gi,g^c=gh,h^a=hij,h^b=hij,h^c=g,i^c=jk,i^h=ik,j^c=ijk,j^g=jk,j^h=jk

Conjugacy classes & character table

118 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 9, 3, F.10*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);