#72229906

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	#13133593 ⋊ C₂
Sylow	C₇ ⋊ #92265

Subgroups

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×AGL(1,ℤ/9)
commutator, cosocle	C₃×C₇×C₉	C₂×C₃²
Frattini	C₃²	C₃×D₃×ΓL(1,𝔽₈)
Fitting	C₃×C₇×3₋¹⁺²	C₂×C₃
socle	C₃²×C₇	C₃²×D₃
prime radical	C₃×C₇×AGL(1,ℤ/9)	C₃
hypocenter	C₇×C₉	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₇×C₉ ▷ C₇×C₉
UCS	C₁ ◁ C₃

142 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 126

Polycyclic presentation:

2,3^4,7,3|b^3=d,e^3=g,b^a=bg^2,c^b=cd^2,e^a=e^2g^2,e^c=eg,f^b=f^4,g^a=g^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#92265, #72230102
maximal normal subgroup of	ΓL(1,𝔽₈)×#92265, #72230102

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