#72247386

order	3240
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₅×C₃²⋊C₉ ⋊ C₂×C₄
Sylow	C₃²⋊C₉ ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #4169⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₅×C₃²⋊C₉	C₂×C₄
Frattini	C₃²	D₃×ΓL(1,𝔽₁₆)
socle	C₃²×C₅	D₃×Dic₃
prime radical	D₅×#4169	C₂
cosocle	D₅×C₃²⋊C₉	C₂²
LCS	G ▷ C₅×C₃²⋊C₉
UCS	C₁

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

normal subgroups only

This group has 220 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) = 90

Polycyclic presentation:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×((C₃⋊D₃)⋊C₉)⋊C₂, #72247468
maximal normal subgroup of	AGL(1,𝔽₅)×((C₃⋊D₃)⋊C₉)⋊C₂, #72247468

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