#72218842

order	6720
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅×C₇×Q₈ ⋊ C₄×D₃
derived	C₅×C₇×SL(2,𝔽₃) ⋊ C₂×C₄
Hall	C₅×C₇ ⋊ C₄×GL(2,𝔽₃)
Sylow	C₇ ⋊ AGL(1,𝔽₅)×GL(2,𝔽₃)
Sylow	C₅ ⋊ #13057073

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72218839
commutator, hypocenter	C₅×C₇×SL(2,𝔽₃)	C₂×C₄
Fitting	C₅×C₇×Q₈	C₄×D₃
socle	C₂×C₅×C₇	C₄×S₄
prime radical	D₅×C₇⋊GL(2,𝔽₃)	C₂
cosocle	C₇×D₅×SL(2,𝔽₃)	C₂²
LCS	G ▷ C₅×C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

352 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 38 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:

2^3,3,5,7,2^3|b^2=c,g^2=i,h^2=i,d^a=d^2,e^b=e^3,e^c=e^4,f^a=f^6,f^b=f^6,g^a=hi,g^d=h,h^a=gi,h^d=ghi,h^g=hi

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 7, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 2, 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.6^6);;
gap> SetConjugate(coll, 7, 1, F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);