#72219099

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ Q₈⋊C₃₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	AΓL(1,𝔽₉)
commutator	#157	C₂×C₃₂
Fitting	C₃²×C₁₆	SD₁₆
socle	C₂×C₃²	Q₈⋊C₁₆
prime radical	C₂×C₃⋊D₃	C₂²⋊C₁₆
cosocle	C₁₆×#157	C₂²
hypocenter	C₃²	Q₈⋊C₃₂
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₁₆

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

normal subgroups only

This group has 135 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

Polycyclic presentation:

2^8,3^2|a^2=c,b^2=f,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,d^b=df,i^a=ij^2,i^b=j^2,i^d=i^2j,i^f=i^2,j^a=j^2,j^b=i,j^d=ij,j^f=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219100, C₃×#72219099, C₃₂×AΓL(1,𝔽₉)
maximal normal subgroup of	C₃×#72219099, C₃₂×AΓL(1,𝔽₉)
maximal quotient of	C₃×#72219099

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