#72217253

order	2520
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃²×C₅×C₇ ⋊ C₂×C₄
Hall	C₃²×C₇ ⋊ C₂×AGL(1,𝔽₅)
Hall	C₃²×C₅ ⋊ C₄×D₇
Hall	C₅×C₇ ⋊ C₂×#157
Sylow	C₃² ⋊ D₇×AGL(1,𝔽₅)
Sylow	C₇ ⋊ D₅⋊#157
Sylow	C₅ ⋊ D₇×#157

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₇	C₂×C₄
prime radical	D₃₅⋊(C₃⋊D₃)	C₂
cosocle	C₇×C₃⋊D₁₅	C₂²
LCS	G ▷ C₃²×C₅×C₇
UCS	C₁

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^2,5,7|a^2=c,d^a=de,d^b=d^2,d^c=d^2,e^a=de^2,e^b=e^2,e^c=e^2,f^a=f^3,f^c=f^4,g^b=g^6

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217254, D₇×D₅⋊#157
maximal normal subgroup of	#72217254, D₇×D₅⋊#157

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