#72248152

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72215875
commutator	#85645	C₂×C₄
Frattini	C₂⁶	C₂×S₄
Fitting	C₂×#15947	D₃
socle	C₂⁵	GL(2,ℤ/4)
prime radical	C₂×#11709314	C₂
cosocle	C₂×#85645	C₂²
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂²⋊C₄
LCS	G ▷ #85645 ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂

6353 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 6353 conjugacy classes of subgroups — too many to draw. Only its 38 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,b^a=bd,e^b=e^2,f^a=fh,f^b=fk,f^d=fk,f^e=g,g^a=gi,g^b=fgjk,g^d=gj,g^e=fg,h^b=hk,h^e=i,i^b=hijk,i^e=hi,j^b=jk,j^e=jk,k^e=j

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248174, #72253856, C₄×#72215875
maximal normal subgroup of	#72248174, #72253856, C₄×#72215875

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