#72269795

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₃³×C₅ ⋊ C₄⋊C₄
Sylow	C₃³ ⋊ C₄⋊AGL(1,𝔽₅)
Sylow	C₅ ⋊ Dic₃⋊(C₃⋊Dic₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊ΓL(1,𝔽₁₆)
commutator, Fitting, socle	C₂×C₃³×C₅	C₂×C₄
prime radical, cosocle	C₂×C₃³×D₅	C₂²
hypocenter	C₃³×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₃³×C₅ ▷ C₃³×C₅
UCS	C₁ ◁ C₂

440 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 180

Polycyclic presentation:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269797, #72269808
maximal normal subgroup of	#72269797, #72269808

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