#72269741

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ #386
Sylow	C₃² ⋊ #65779
Sylow	C₅ ⋊ #10593453

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₅)×GO⁺(4,𝔽₂)
commutator	C₂×C₅×C₃⋊D₃	C₂²×C₄
Fitting, socle	C₂×C₃²×C₅	C₄×D₄
prime radical	C₂×D₅×GO⁺(4,𝔽₂)	C₂
cosocle	C₂×D₅×C₃⋊D₃	C₂³
hypocenter	C₃²×C₅	#386
LCS	G ▷ C₂×C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

940 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 60

Polycyclic presentation:

2^6,3^2,5|a^2=d,b^a=bf,c^b=ce,g^a=gh,g^b=g^2,g^c=gh,g^e=g^2,g^f=g^2,h^a=h^2,h^b=g^2h,h^c=h^2,h^e=h^2,h^f=h^2,i^a=i^3,i^d=i^4

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#64241
maximal normal subgroup of	AGL(1,𝔽₅)×#64241

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);