#72269216

order	3000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₅³ ⋊ C₂×C₄
Sylow	C₅³ ⋊ C₄×D₃
Sylow	C₃ ⋊ D₅×#1042

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₅³	C₂×C₄
prime radical	C₅×D₅×D₁₅	C₂
cosocle	C₃×C₅²×D₅	C₂²
LCS	G ▷ C₃×C₅³
UCS	C₁

176 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₁₅⋊D₅
maximal normal subgroup of	AGL(1,𝔽₅)×D₁₅⋊D₅

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