#72217838

order	3360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×C₅×C₇ ⋊ C₃×C₄
Hall	#11765360 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ #117
Hall	C₅×C₇ ⋊ C₃×#117
Hall	ΓL(1,𝔽₈) ⋊ #4130
Sylow	C₇ ⋊ C₃×#4130
Sylow	C₅ ⋊ #6489⋊C₃

Subgroups

	subgroup	G/H
center	C₄	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₇	C₂²×C₃×C₄
Frattini	C₂	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₇	C₃×C₄
prime radical	C₂²×D₅×ΓL(1,𝔽₈)	C₂²
cosocle	C₂×C₇×D₅	C₂³×C₃
hypercenter	C₂×C₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₃×#117
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

304 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 97 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,c^2=f,b^a=bf,c^a=cf,g^a=g^3,g^e=g^4,h^a=h^6,h^d=h^2

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218128, #72218254, #72218319, …
maximal normal subgroup of	#72218128, #72218254, #72218319, …

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, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^3);;
gap> SetConjugate(coll, 7, 5, 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);