#72258367

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#13794 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#85547
commutator, cosocle, hypocenter	#13794⋊C₃	C₂²
Frattini, socle	C₂⁴	C₂×#1020
Fitting	C₂×#13794	D₃
hypercenter	C₂³	#85547
LCS	G ▷ #13794⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂³

6186 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 6186 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 96

Polycyclic presentation:

2^2,3,2^8|d^2=i,e^2=ij,b^a=bk,c^a=c^2,c^b=cijk,d^a=ejk,d^c=eik,e^a=djk,e^c=dek,e^d=ek,f^c=g,f^d=fh,f^e=fh,g^a=fg,g^c=fg,g^e=gh,i^a=ij,i^c=ij,j^c=ik

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72258531

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