#72213709

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₃²×C₅ ⋊ GL(2,𝔽₃)
derived	C₅×PSU(3,𝔽₂)⋊C₃ ⋊ C₂
Sylow	C₅ ⋊ AGL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×PSU(3,𝔽₂)⋊C₃	C₂
Fitting, socle	C₃²×C₅	GL(2,𝔽₃)
LCS	G ▷ C₅×PSU(3,𝔽₂)⋊C₃
UCS	C₁

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,5,2^3,3^2|d^2=f,e^2=f,b^a=b^2,c^a=c^4,d^a=de,d^b=e,e^a=ef,e^b=def,e^d=ef,g^a=g^2,g^d=h,g^e=gh,g^f=g^2,h^a=g^2h,h^b=gh,h^d=g^2,h^e=gh^2,h^f=h^2

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×AGL(2,𝔽₃), C₃×#72213709, C₂×#72213709
maximal normal subgroup of	D₅×AGL(2,𝔽₃), C₃×#72213709, C₂×#72213709
maximal quotient of	C₃×#72213709, C₂×#72213709

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