#72218232

order	3360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	#11764798 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ AGL(1,ℤ/8)
Hall	C₅×C₇ ⋊ C₃×AGL(1,ℤ/8)
Hall	ΓL(1,𝔽₈) ⋊ #4095
Sylow	C₇ ⋊ C₃×#4095
Sylow	C₅ ⋊ #6495⋊C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#63305⋊C₃
commutator, cosocle	C₄×C₅×C₇	C₂³×C₃
Frattini	C₄	C₂×D₅×AGL(1,𝔽₇)
Fitting	C₂×C₄×C₅×C₇	C₂²×C₃
socle	C₂×C₅×C₇	C₂×C₃×D₄
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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|a^2=e,e^2=f,b^a=be,c^a=cf,e^b=ef,g^a=g^4,g^b=g^4,h^a=h^6,h^d=h^2

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#6495⋊C₃, #72218484, AGL(1,𝔽₇)×#4095
maximal normal subgroup of	D₅×#6495⋊C₃, #72218484, AGL(1,𝔽₇)×#4095

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