#72218535

order	6720
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₇×D₄ ⋊ C₂²×C₃
Hall	#72218534 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ C₂×#120
Hall	C₅×C₇ ⋊ C₂×C₃×#120
Hall	ΓL(1,𝔽₈) ⋊ D₅×#120
Sylow	C₇ ⋊ C₃×D₅×#120
Sylow	C₅ ⋊ #120×AGL(1,𝔽₇)

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₅×C₇	C₂⁴×C₃
Fitting	C₂×C₅×C₇×D₄	C₂²×C₃
hypercenter	C₂×D₄	D₅×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₂×C₃×#120
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

2648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^2,5,7|c^2=f,c^a=cf,c^b=cfg,h^b=h^4,h^d=h^4,i^d=i^6,i^e=i^2

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#120×AGL(1,𝔽₇)
maximal normal subgroup of	D₅×#120×AGL(1,𝔽₇)

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