#72218554

order	4480
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅×C₇×M₁₆ ⋊ C₂×C₄
Hall	C₅×C₇ ⋊ C₄×AGL(1,ℤ/8)
Sylow	C₇ ⋊ AGL(1,𝔽₅)×AGL(1,ℤ/8)
Sylow	C₅ ⋊ #11736644

Subgroups

	subgroup	G/H
center	C₂	C₂×#11765324
commutator	C₄×C₅×C₇	C₂³×C₄
Frattini	C₄	C₂²×D₇×AGL(1,𝔽₅)
Fitting	C₅×C₇×M₁₆	C₂×C₄
socle	C₂×C₅×C₇	C₂×C₄×D₄
prime radical	D₅×#6471	C₂
cosocle	C₄×C₇×D₅	C₂⁴
hypercenter	M₁₆	D₇×AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

1096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5,7|a^2=f,d^2=e,f^2=g,b^a=bf,c^a=cg,f^b=fg,h^d=h^3,h^e=h^4,i^b=i^6,i^d=i^6

Conjugacy classes & character table

151 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218556, D₇×AGL(1,𝔽₅)×AGL(1,ℤ/8)
maximal normal subgroup of	#72218556, D₇×AGL(1,𝔽₅)×AGL(1,ℤ/8)

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