#72222379

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃³×C₅ ⋊ C₂²×C₄
Sylow	C₃³ ⋊ C₂²×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #88949

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³×C₅	C₂²×C₄
prime radical	D₅×#1113	C₂
cosocle	C₃³×D₅	C₂³
LCS	G ▷ C₃³×C₅
UCS	C₁

540 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃³×AGL(1,𝔽₅), #72222606, #72254914, …
maximal normal subgroup of	D₃³×AGL(1,𝔽₅), C₂×#72222379
maximal quotient of	#72222606, #72254802, #72254914, …

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