#72242191

order	2560
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1368 ⋊ C₄
Sylow	C₅ ⋊ #94057

Subgroups

	subgroup	G/H
center	C₂	#13013594
commutator	C₅×C₁₆	C₄×C₈
Frattini	#346	C₂×AGL(1,𝔽₅)
Fitting	C₅×#1368	C₄
socle	C₂×C₅	#7178
prime radical	C₂²×D₅	#360
cosocle	D₅×#346	C₂²
hypercenter	#1368	AGL(1,𝔽₅)
hypocenter	C₅	#94057
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1368

308 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 160

Polycyclic presentation:

2^9,5|a^2=c,b^2=cdghi,c^2=f,d^2=g,e^2=gi,f^2=h,g^2=h,h^2=i,b^a=be,c^b=cg,d^a=dhi,d^b=dg,d^c=di,e^a=eh,e^b=egi,e^c=ei,f^b=fh,g^a=gi,g^b=gh,h^b=hi,j^a=j^3,j^c=j^4,j^d=j^4

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#7389, #72242252, #72242283
maximal normal subgroup of	AGL(1,𝔽₅)×#7389, #72242252, #72242283

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, 2, 2, 2, 2, 2, 2, 2, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.4*F.7*F.8*F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^3);;
gap> SetConjugate(coll, 10, 3, F.10^4);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);