#72269161

order	3200
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1398

Subgroups

	subgroup	G/H
center	C₈	#87830
commutator	C₄×C₅²	C₄×C₈
Frattini, hypercenter	C₁₆	D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×C₁₆	C₂×C₄
socle	C₂×C₅²	#346
prime radical	C₂×C₅⋊D₅	M₃₂
cosocle	C₁₆×C₅⋊D₅	C₂²
hypocenter	C₅²	#1398
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₁₆

174 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 43 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:

2^7,5^2|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,b^a=bf,c^a=cg,d^b=dg,h^a=h^4,h^b=h^3,h^c=h^4,i^a=h^2i,i^b=i^3,i^c=i^4

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269166
maximal normal subgroup of	#72269166

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