#72253196

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#99 ⋊ D₄
Sylow	C₃² ⋊ #24404

Subgroups

	subgroup	G/H
center	C₂	#11809569
commutator, cosocle	C₂³×C₃⋊D₃	C₂⁴
Frattini	C₂³	C₂²×GO⁺(4,𝔽₂)
Fitting	C₃²×#99	D₄
socle	C₂×C₃²	#2410
hypercenter	#99	GO⁺(4,𝔽₂)
hypocenter	C₃²	#24404
LCS	G ▷ C₂³×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #99

3075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^8,3^2|b^2=g,c^2=gh,c^a=cf,c^b=ce,d^b=dgh,d^c=dgh,e^a=eh,f^b=fh,i^b=ij,i^c=ij,i^d=ij,i^g=i^2,j^b=ij^2,j^c=ij^2,j^d=j^2,j^g=j^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×GO⁺(4,𝔽₂)
maximal normal subgroup of	B(4,𝔽₂)×GO⁺(4,𝔽₂)

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