#72242745

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×H(5,𝔽₂) ⋊ Q₈
Sylow	C₃² ⋊ #32975

Subgroups

	subgroup	G/H
center	C₂	#11856243
commutator, cosocle	C₂³×C₃⋊D₃	C₂⁴
Frattini	C₂³	C₂²×PSU(3,𝔽₂)
Fitting	C₃²×H(5,𝔽₂)	Q₈
socle	C₂×C₃²	#2848
prime radical	C₃⋊D₃×H(5,𝔽₂)	C₂²
hypercenter	H(5,𝔽₂)	PSU(3,𝔽₂)
hypocenter	C₃²	#32975
LCS	G ▷ C₂³×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

1999 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1999 conjugacy classes of subgroups — too many to draw. Only its 165 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×PSU(3,𝔽₂), #72253242, #72253243
maximal normal subgroup of	B(4,𝔽₂)×PSU(3,𝔽₂), #72253242, #72253243

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