#72259757

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₄×SD₁₆

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×#89092
commutator	C₃²×#157	C₂²×C₄
Fitting, socle	C₂×C₃⁴	C₂×SD₁₆
prime radical	C₂×#157⋊(C₃⋊D₃)	C₂²
cosocle	C₂×C₃²×#157	C₂³
hypocenter	C₃⁴	C₄×SD₁₆
LCS	G ▷ C₃²×#157 ▷ C₃²×C₃⋊D₃ ▷ C₃⁴
UCS	C₁ ◁ C₂

769 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 769 conjugacy classes of subgroups — too many to draw. Only its 90 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:

2^6,3^4|a^2=d,c^2=f,e^2=f,c^b=ce,e^b=ef,e^c=ef,g^a=g^2,g^b=g^2,h^a=g^2h,h^b=h^2,i^b=i^2,i^c=ij,i^e=i^2j,i^f=i^2,j^b=ij,j^c=ij^2,j^e=ij,j^f=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72259802

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