#72246782

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃²×C₇ ⋊ C₂²
Hall	C₃²×C₇ ⋊ C₂×#120
Sylow	C₃² ⋊ D₇×#120
Sylow	C₇ ⋊ C₃⋊D₃×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²×C₇	C₂⁴
Fitting	C₂⁴×C₃²×C₇	C₂²
hypercenter	C₂⁴	D₇×C₃⋊D₃
hypocenter	C₃²×C₇	C₂×#120
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂⁴

3972 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

normal subgroups only

This group has 3972 conjugacy classes of subgroups — too many to draw. Only its 336 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) = 504

Polycyclic presentation:

2^6,3^2,7|b^a=be,d^a=df,g^a=g^2,g^c=g^2,h^a=h^2,h^c=h^2,i^c=i^6

Conjugacy classes & character table

318 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×C₃⋊D₃×#120
maximal normal subgroup of	D₇×C₃⋊D₃×#120

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