#72241321

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72240149 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂²×#85645
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₂³×C₃
Frattini, socle	C₂³	C₂³×C₂²⋊A₄
Fitting	#72240149	C₃
hypercenter	D₄	#85645
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂ ◁ D₄

At least 6440 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has at least 6440 conjugacy classes of subgroups — too many to draw. Only its 157 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:

2^3,3,2^7|b^2=k,b^a=bk,e^d=ef,f^d=e,g^c=gi,g^d=gh,g^f=gk,h^c=hj,h^d=g,h^e=hk,i^d=ij,j^d=i

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241363
maximal normal subgroup of	#72241363
maximal quotient of	#72267166

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