#72248147

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

derived

C₂²⋊(C₂²⋊A₄) ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209642
commutator, hypocenter	C₂²⋊(C₂²⋊A₄)	C₂²×C₄
Frattini, socle	C₂⁵	C₂²×S₄
Fitting	C₂×#15947	D₃
prime radical	C₂×#11709314	C₂
cosocle	C₂×C₂²⋊(C₂²⋊A₄)	C₂³
LCS	G ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂

9659 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 9659 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 48

Polycyclic presentation:

2^4,3,2^6|b^2=d,e^b=e^2,f^a=fh,f^c=fk,f^e=g,g^a=gi,g^b=fg,g^c=gj,g^e=fg,h^e=i,i^b=hi,i^e=hi,j^b=jk,j^e=jk,k^e=j

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248174, C₄×#72209642, #72253884
maximal normal subgroup of	#72248174, C₄×#72209642, #72253884

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.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, 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);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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.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);