#72249184

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇×C₃²⋊C₉ ⋊ C₂×C₃
Hall	#13133545 ⋊ C₂
Sylow	C₇ ⋊ #92302

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊#67361
commutator, Fitting, cosocle, hypocenter	C₇×C₃²⋊C₉	C₂×C₃
Frattini	C₃²	C₃⋊D₃×ΓL(1,𝔽₈)
socle	C₃²×C₇	C₃×C₃⋊D₃
LCS	G ▷ C₇×C₃²⋊C₉
UCS	C₁ ◁ C₃

294 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 36 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:

2,3^3,7,3^2|c^3=g,b^a=bg^2,c^a=c^2g^2,c^b=cg,d^a=d^2,d^c=df,e^b=e^2,g^a=g^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#92302, #72269733
maximal normal subgroup of	ΓL(1,𝔽₈)×#92302, #72269733

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