#72218301

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

Aliases

No aliases for this group.

Splittings

Hall	#11765320 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ C₂×M₁₆
Hall	C₅×C₇ ⋊ C₂×C₃×M₁₆
Hall	ΓL(1,𝔽₈) ⋊ D₅⋊M₁₆
Sylow	C₇ ⋊ C₃×D₅⋊M₁₆
Sylow	C₅ ⋊ M₁₆×AGL(1,𝔽₇)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇)
commutator, socle	C₂×C₅×C₇	C₂²×C₃×C₄
Fitting	C₄×C₅×C₇	C₂×C₃×C₄
prime radical	C₂×D₅×AGL(1,𝔽₇)	C₄
cosocle	C₇×Dic₅	C₂³×C₃
hypercenter	C₄	AGL(1,𝔽₅)×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄

272 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 84 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,b^2=f,c^2=f,e^2=f,b^a=bf,g^a=g^3,g^e=g^4,h^c=h^6,h^d=h^2

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×D₅⋊M₁₆
maximal normal subgroup of	AGL(1,𝔽₇)×D₅⋊M₁₆

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