#72218345

order	6720
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅×C₇ ⋊ C₃×C₄
Hall	#72218343 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ #379
Hall	C₅×C₇ ⋊ C₃×#379
Hall	ΓL(1,𝔽₈) ⋊ #65800
Sylow	C₇ ⋊ C₃×#65800
Sylow	C₅ ⋊ #13052147

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅×C₇	C₂²×C₃×C₄
Fitting	C₂⁴×C₅×C₇	C₃×C₄
prime radical	C₂⁴×D₅×ΓL(1,𝔽₈)	C₂
cosocle	C₂²×C₇×D₅	C₂³×C₃
hypercenter	C₂⁴	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₃×#379
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂⁴

2024 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 2024 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×#10620789, AGL(1,𝔽₇)×#65800
maximal normal subgroup of	#120×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×#10620789, AGL(1,𝔽₇)×#65800

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