#72253242

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×H(5,𝔽₂) ⋊ SD₁₆
Sylow	C₃² ⋊ #346526

Subgroups

	subgroup	G/H
center	C₂	#72220542
commutator, cosocle	C₂³×#157	C₂⁴
Frattini	C₂³	C₂²×AΓL(1,𝔽₉)
Fitting	C₃²×H(5,𝔽₂)	SD₁₆
socle	C₂×C₃²	#32881
prime radical	H(5,𝔽₂)×GO⁺(4,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	AΓL(1,𝔽₉)
hypocenter	C₃²	#346526
LCS	G ▷ C₂³×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

10676 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 10676 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 72

Polycyclic presentation:

2^9,3^2|b^2=h,c^2=i,g^2=i,c^a=cf,c^b=ce,d^c=dg,e^a=eh,f^b=fh,g^c=gi,g^d=gi,j^c=jk,j^g=k,j^i=j^2,k^c=jk^2,k^d=k^2,k^g=j^2,k^i=k^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×AΓL(1,𝔽₉)
maximal normal subgroup of	B(4,𝔽₂)×AΓL(1,𝔽₉)

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