#72341421

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁶×D₄ ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂²×#1020
commutator, cosocle	C₂³×C₂²⋊A₄	C₂³
Fitting	C₂⁶×D₄	D₃
socle	C₂⁷	C₂²×D₃
hypercenter	C₂²×D₄	#1020
hypocenter	C₂²⋊A₄	#392
LCS	G ▷ C₂³×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

At least 7597 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has at least 7597 conjugacy classes of subgroups — too many to draw. Only its 173 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:

2^6,3,2^4|b^2=f,b^a=bf,c^a=ce,c^b=cd,g^b=g^2,h^b=i,h^g=hi,i^b=h,i^g=h,j^b=jk,j^g=k,k^g=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341433
maximal normal subgroup of	#72341433

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