#72218372

order	3360
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×C₅×C₇ ⋊ C₂²×C₃
Hall	#11765387 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ C₂×#41
Hall	C₅×C₇ ⋊ C₂×C₃×#41
Hall	ΓL(1,𝔽₈) ⋊ D₅×#41
Sylow	C₇ ⋊ C₃×D₅×#41
Sylow	C₅ ⋊ #41×AGL(1,𝔽₇)

Subgroups

	subgroup	G/H
center	C₄	C₂×D₅×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₅×C₇	C₂⁴×C₃
Frattini	C₂	C₂²×D₅×AGL(1,𝔽₇)
Fitting	C₂×C₄×C₅×C₇	C₂²×C₃
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₂×C₃×#41
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#41×AGL(1,𝔽₇), #72218484, #72218497
maximal normal subgroup of	D₅×#41×AGL(1,𝔽₇), #72218484, #72218497

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