#72268929

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²	#72218804
commutator	C₂²×#5948	C₂×C₄
Frattini	C₂³×C₂²⋊C₄	C₂×S₄
Fitting	#72268824	D₃
socle	C₂⁴	#73145
prime radical	#72268831	C₂
cosocle	C₂×#14885⋊C₃	C₂²
hypercenter	C₂²×D₄	#4625
hypocenter	C₂²⋊SL(2,𝔽₃)	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂²×#5948 ▷ C₂²×C₂²⋊SL(2,𝔽₃) ▷ C₂×C₂²⋊SL(2,𝔽₃) ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

37854 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 37854 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 192

Polycyclic presentation:

2^6,3,2^5|b^2=c,d^2=l,h^2=l,i^2=l,b^a=bd,c^a=ce,d^a=dl,d^b=del,d^c=df,e^b=ef,g^b=g^2,h^b=hij,h^c=hj,h^g=i,i^b=ikl,i^c=ijk,i^g=hil,i^h=il,j^b=k,j^g=jk,k^b=j,k^g=j

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72268932

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