#72249201

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

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₇×#6909 ⋊ C₂
derived	C₇×AΣL(1,𝔽₂₇) ⋊ C₂×C₃
Sylow	#6909 ⋊ D₇
Sylow	C₇ ⋊ #4183⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×AΣL(1,𝔽₂₇)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₂₁
Fitting	C₇×#6909	C₂
socle	C₃×C₇	C₃×#272
LCS	G ▷ C₇×AΣL(1,𝔽₂₇)
UCS	C₁

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

normal subgroups only

This group has 200 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) = 189

Polycyclic presentation:

2,3^3,7,3^2|b^a=bg,c^a=c^2,c^b=cg,d^a=d^2,d^b=dg,d^c=df,e^a=e^6,f^a=fg^2,f^d=fg,g^a=g^2

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249205, #72254880, D₇×#4183⋊C₃
maximal normal subgroup of	#72254880, D₇×#4183⋊C₃

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