#72231064

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ C₂×D₃
derived	C₃×C₅×#997 ⋊ C₂²
Sylow	C₅ ⋊ D₃×#5935

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72231058
commutator, cosocle, hypocenter	C₃×C₅×#997	C₂²
Fitting	C₃×C₅×H(5,𝔽₂)	C₂×D₃
socle	C₂×C₃×C₅	C₂×#1020
LCS	G ▷ C₃×C₅×#997
UCS	C₁ ◁ C₂

1368 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1368 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 generators on 16 points:

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

Polycyclic presentation:

2^2,3^2,5,2^5|f^2=j,g^2=j,c^a=c^2,d^b=d^2,e^a=e^4,e^b=e^4,f^a=fg,f^c=g,g^a=gj,g^c=fgj,g^f=gj,h^a=hi,h^c=i,h^g=hj,i^c=hi,i^f=ij

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₅×#5935
maximal normal subgroup of	D₃×D₅×#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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 5, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);