#72230913

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ D₃
derived	C₃×C₅×#997 ⋊ C₂
Sylow	C₅ ⋊ C₃⋊#5935

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊(C₅⋊#1020)
commutator, cosocle, hypocenter	C₃×C₅×#997	C₂
Fitting	C₃×C₅×H(5,𝔽₂)	D₃
socle	C₂×C₃×C₅	#1020
LCS	G ▷ C₃×C₅×#997
UCS	C₁ ◁ C₂

384 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 16 points:

(1,3,4,7)(2,6)(5,8)(10,11)(12,13)(15,16);(9,10,12,13,11)(14,15,16);(2,3,8)(5,6,7)

Polycyclic presentation:

2,3^2,5,2^5|e^2=i,f^2=i,b^a=b^2,c^a=c^2,d^a=d^4,e^a=ef,e^b=f,f^a=fi,f^b=efi,f^e=fi,g^a=gh,g^b=h,g^f=gi,h^b=gh,h^e=hi

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×C₃⋊#5935, D₁₅×#5935, D₃×C₅⋊#5935
maximal normal subgroup of	D₅×C₃⋊#5935, D₁₅×#5935, D₃×C₅⋊#5935

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