#72246809

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ C₂×#100
Sylow	C₃² ⋊ C₂×#6397
Sylow	C₇ ⋊ #10591503

Subgroups

	subgroup	G/H
center	C₂	#72246808
commutator	C₂²×C₃²×C₇	C₂²×C₄
Frattini, hypercenter	C₂×C₄	D₃²×D₇
Fitting, cosocle	C₂×C₃²×C₄×C₇	C₂³
socle	C₂×C₃²×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₃⋊D₃×D₂₈	C₂
hypocenter	C₃²×C₇	C₂×#100
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

980 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 168

Polycyclic presentation:

2^6,3^2,7|a^2=d,d^2=f,e^2=f,b^a=bdef,d^b=df,e^a=ef,e^b=ef,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^b=i^6

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#10591503, #72246878, #72246957
maximal normal subgroup of	D₇×#10591503, #72246878, #72246957

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);