#72340793

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#3549 ⋊ D₃
derived	C₂²⋊#5464 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²⋊#5464	C₂²
Frattini	C₂²	C₂×#87588
Fitting	C₂²×#3549	D₃
socle	C₂⁴	C₂×#1020
LCS	G ▷ C₂²⋊#5464
UCS	C₁

11077 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 11077 conjugacy classes of subgroups — too many to draw. Only its 40 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:

2^2,3,2^8|a^2=jk,h^2=k,i^2=j,c^a=c^2k,d^a=dej,d^b=dk,d^c=e,e^a=ejk,e^b=ejk,e^c=de,f^a=fg,f^c=g,g^c=fg,h^a=ijk,h^c=hi,h^d=hk,h^e=hj,i^a=hjk,i^c=hj,i^d=ijk,i^e=ik,j^a=k,j^c=k,k^a=j,k^c=jk

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340794, #72340795, #72340799, …
maximal normal subgroup of	#72340794, #72340795, #72340799, …

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