#72222342

order	2430
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×#6900 ⋊ C₂
derived	C₅×C₉² ⋊ C₂×C₃
Sylow	#6900 ⋊ D₅
Sylow	C₅ ⋊ #92314

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×C₉²	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₁₅
Fitting	C₅×#6900	C₂
socle	C₃²×C₅	C₃×C₃⋊D₃
LCS	G ▷ C₅×C₉²
UCS	C₁

208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#92314, #72222373
maximal normal subgroup of	D₅×#92314, #72222373

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