#72235028

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	H(5,𝔽₄) ⋊ C₃
derived	C₂²×H(3,𝔽₄) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×H(3,𝔽₄)	C₂²×C₃
Frattini, socle	C₂²	C₂²×C₂²⋊(C₂²⋊A₄)
Fitting	H(5,𝔽₄)	C₃
LCS	G ▷ C₂²×H(3,𝔽₄)
UCS	C₁

143951 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 143951 conjugacy classes of subgroups — too many to draw. Only its 226 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|d^2=j,e^2=k,c^a=cjk,c^b=cjk,d^c=ej,e^c=de,f^a=fjk,f^b=fj,f^c=fg,f^d=fj,f^e=fjk,g^a=gk,g^b=gjk,g^c=f,g^d=gk,g^e=gj,h^c=i,h^d=hk,h^e=hj,i^c=hi,i^d=ij,i^e=ijk,j^c=k,k^c=jk

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72235031

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