#72222352

order	4860
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×#6900 ⋊ C₄
derived	C₅×C₉² ⋊ C₃×C₄
Sylow	#6900 ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #11757618

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×C₉²	C₃×C₄
Frattini	C₃²	C₃×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₅×#6900	C₄
socle	C₃²×C₅	C₃×C₃⋊Dic₃
prime radical	C₃³×D₅	C₃⋊D₃
cosocle	C₉²×D₅	C₂×C₃
LCS	G ▷ C₅×C₉²
UCS	C₁

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2,3^2,5,3^2|a^2=c,d^3=g,e^3=h,d^a=d^2g^2,d^b=dg^2,e^a=e^2h^2,e^b=eh^2,f^a=f^3,f^c=f^4,g^a=g^2,h^a=h^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#92314, #72222406
maximal normal subgroup of	AGL(1,𝔽₅)×#92314, #72222406

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