#72249197

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₇×C₃²⋊H(3,𝔽₃) ⋊ C₂
Sylow	C₃²⋊H(3,𝔽₃) ⋊ D₇
Sylow	C₇ ⋊ #92338

Subgroups

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

534 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 75 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=b^2,c^a=c^2,c^b=cg,d^a=d^2,d^c=df,e^a=e^6

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254918, D₇×#92338
maximal normal subgroup of	#72254918, D₇×#92338
maximal quotient of	#72249205

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