#72209686

order	3000
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	H(3,𝔽₅) ⋊ SL(2,𝔽₃)
derived, Hall	#11758708 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₅	#6057⋊C₃
commutator, cosocle, hypocenter	#11758708	C₃
Fitting	H(3,𝔽₅)	SL(2,𝔽₃)
LCS	G ▷ #11758708
UCS	C₁ ◁ C₅

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^3,5^3|b^2=d,c^2=d,b^a=c,c^a=bcd,c^b=cd,e^a=e^3fg,e^b=fg^4,e^c=f^2g^4,e^d=e^4g^3,f^a=e^2fg^3,f^b=e^4g^4,f^c=e^2g^2,f^d=f^4,f^e=fg

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72209687

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