#72259504

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊PSU(3,𝔽₂)
commutator, cosocle	C₂×C₃²×C₃⋊D₃	C₂³
Fitting, socle	C₂×C₃⁴	C₂×Q₈
prime radical	C₂×C₃×D₃×C₃⋊D₃	C₂²
hypocenter	C₃⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₃²×C₃⋊D₃ ▷ C₃⁴
UCS	C₁ ◁ C₂

457 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 66 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:

2^5,3^4|b^2=e,c^2=e,b^a=bd,c^b=ce,f^a=f^2,f^b=f^2,g^a=f^2g,g^b=g^2,h^b=h^2i^2,h^c=hi^2,h^e=h^2,i^b=h^2i,i^c=h^2i^2,i^e=i^2

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#626×PSU(3,𝔽₂)
maximal normal subgroup of	#626×PSU(3,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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);