#72218361

order	3360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₅×C₇ ⋊ C₂²×C₃
Hall	#11765025 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ #120
Hall	C₅×C₇ ⋊ C₃×#120
Hall	ΓL(1,𝔽₈) ⋊ #4083
Sylow	C₇ ⋊ C₃×#4083
Sylow	C₅ ⋊ #6500⋊C₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³×C₃
Fitting	C₂³×C₅×C₇	C₂²×C₃
hypercenter	C₂³	D₅×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₃×#120
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

520 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 91 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:

2^3,3,2^2,5,7|c^2=e,c^a=ce,c^b=cef,g^a=g^4,g^c=g^4,h^b=h^6,h^d=h^2

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#6500⋊C₃, AGL(1,𝔽₇)×#4083, #72218535
maximal normal subgroup of	D₅×#6500⋊C₃, AGL(1,𝔽₇)×#4083, #72218535

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