#72261972

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂	C₂×#11714268
commutator	C₂×C₂²⋊(C₄²⋊C₃)	C₂×C₄
Frattini	C₂⁴	C₂×#1020
Fitting	C₂⁴×#127	D₃
socle	C₂⁵	C₂×SL(2,ℤ/4)
prime radical	C₂²×#85647	C₂
cosocle	C₂×#85647	C₂²
hypercenter	C₂²	#11714268
hypocenter	C₂²⋊(C₄²⋊C₃)	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊(C₄²⋊C₃) ▷ C₂²⋊(C₄²⋊C₃)
UCS	C₁ ◁ C₂ ◁ C₂²

23711 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 23711 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 24

Polycyclic presentation:

2^4,3,2^6|b^2=c,f^2=j,g^2=k,b^a=bd,e^b=e^2,f^b=gk,f^c=fj,f^e=gj,g^b=f,g^c=gk,g^e=fg,h^e=i,i^b=hi,i^e=hi,j^b=k,j^e=k,k^b=j,k^e=jk

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72262094

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