#72249205

order	10206
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₇×#10624749 ⋊ C₂
Sylow	#10624749 ⋊ D₇
Sylow	C₇ ⋊ #13091120

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×#10624749	C₂
Frattini	C₃³	C₃⋊(C₃⋊D₂₁)
socle	C₃×C₇	#92338
LCS	G ▷ C₇×#10624749
UCS	C₁

700 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 77 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:

2,3^3,7,3^3|d^3=h,b^a=b^2,c^a=c^2,c^b=cg,d^a=d^2h^2,d^c=df,e^a=e^6,f^a=fh^2,f^b=fh,f^d=fh,g^d=gh^2,h^a=h^2

Conjugacy classes & character table

213 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254920, D₇×#13091120
maximal normal subgroup of	#72254920, D₇×#13091120

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", "h");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 7, 3, 3, 3]);;
gap> SetPower(coll, 4, F.8);;
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*F.8^2);;
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.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);