#72222979

order	10368
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₃² ⋊ C₂×D₃²
derived	C₃²×A₄² ⋊ C₂³

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×A₄²	C₂³
Fitting, socle	C₂⁴×C₃²	C₂×D₃²
LCS	G ▷ C₃²×A₄²
UCS	C₁

6741 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 6741 conjugacy classes of subgroups — too many to draw. Only its 88 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:

2^3,3^4,2^4|d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^2,f^c=f^2,g^a=f^2g,g^c=g^2,h^a=i,h^b=i,h^d=hi,h^e=i,i^a=h,i^b=h,i^d=h,i^e=hi,j^a=hij,j^b=hk,j^d=ij,j^e=jk,k^b=ij,k^d=hk,k^e=j

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72226353

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