#72217686

order	3360
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₅×C₇×Q₈ ⋊ C₃×C₄
Hall	Q₈×C₇⋊AGL(1,𝔽₅) ⋊ C₃
Hall	C₅×C₇ ⋊ C₄×SL(2,𝔽₃)
Sylow	C₇ ⋊ AGL(1,𝔽₅)×SL(2,𝔽₃)
Sylow	C₅ ⋊ Dic₇⋊SL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	(C₂²⋊ΓL(1,𝔽₈))⋊AGL(1,𝔽₅)
commutator, Fitting, hypocenter	C₅×C₇×Q₈	C₃×C₄
socle	C₂×C₅×C₇	C₄×A₄
prime radical	D₅×C₇⋊SL(2,𝔽₃)	C₂
cosocle	C₇×Q₈×D₅	C₂×C₃
LCS	G ▷ C₅×C₇×Q₈
UCS	C₁ ◁ C₂

120 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 28 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|a^2=c,d^2=h,e^2=h,d^b=de,e^b=d,e^d=eh,f^a=f^3,f^c=f^4,g^a=g^6,g^b=g^2

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SL(2,𝔽₃)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×D₇⋊SL(2,𝔽₃)
maximal normal subgroup of	SL(2,𝔽₃)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×D₇⋊SL(2,𝔽₃)

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