#72249200

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₇×C₃²⋊H(3,𝔽₃) ⋊ C₂
derived	C₃×C₇×H(3,𝔽₃) ⋊ C₂×C₃
Sylow	C₃²⋊H(3,𝔽₃) ⋊ D₇
Sylow	C₇ ⋊ #92300

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊#67385
commutator, cosocle, hypocenter	C₃×C₇×H(3,𝔽₃)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₂₁
Fitting	C₇×C₃²⋊H(3,𝔽₃)	C₂
socle	C₃²×C₇	C₃×C₃⋊D₃
LCS	G ▷ C₃×C₇×H(3,𝔽₃)
UCS	C₁ ◁ C₃

384 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 44 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|c^a=c^2,c^b=ce,d^a=d^2,d^c=dg,e^a=e^2,f^a=f^6

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249205, #72254872, D₇×#92300
maximal normal subgroup of	#72254872, D₇×#92300

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