#72218641

order	5040
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₅×C₇ ⋊ C₃×Dic₃
derived	C₅×C₇×A₄ ⋊ C₃×C₄
Hall	C₅×C₇ ⋊ C₃×SL(2,ℤ/4)
Sylow	C₇ ⋊ C₃×A₄⋊AGL(1,𝔽₅)
Sylow	C₅ ⋊ A₄⋊(ΓL(1,𝔽₈)⋊C₄)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅×C₇×A₄	C₃×C₄
Fitting, socle	C₂²×C₅×C₇	C₃×Dic₃
prime radical	D₅×A₄×ΓL(1,𝔽₈)	C₂
cosocle	C₇×D₅×A₄	C₂×C₃
LCS	G ▷ C₅×C₇×A₄
UCS	C₁

168 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 29 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:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×A₄⋊AGL(1,𝔽₇), AGL(1,𝔽₇)×A₄⋊AGL(1,𝔽₅)
maximal normal subgroup of	S₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×A₄⋊AGL(1,𝔽₇), AGL(1,𝔽₇)×A₄⋊AGL(1,𝔽₅)
maximal quotient of	#72218642

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