#72256228

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×#379 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#87588⋊C₂
commutator	C₂×C₂²⋊(C₂²⋊A₄)	C₂×C₄
Frattini	C₂⁴	C₂×#1020
Fitting	C₂³×#379	D₃
socle	C₂⁶	C₂×S₄
prime radical	C₂²×#87588	C₂
cosocle	C₂²×C₂²⋊(C₂²⋊A₄)	C₂²
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂²

At least 14910 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has at least 14910 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^6|a^2=d,b^2=cd,b^a=bc,e^a=e^2,f^a=fgh,f^b=fj,f^e=g,g^a=gi,g^b=gk,g^e=fg,h^a=hi,h^e=hi,i^e=h,j^a=jk,j^e=k,k^e=jk

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256263, #72256321, #72256454
maximal normal subgroup of	#72256263, #72256321, #72256454

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