#72242270

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #2177

Subgroups

	subgroup	G/H
center	C₂²	D₈×C₅⋊D₅
commutator	C₅²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₅⋊D₅
Fitting, prime radical	C₂×C₅²×D₁₆	C₂
socle	C₂²×C₅²	C₂×D₈
cosocle	C₂×C₅²×C₈	C₂³
hypercenter	C₂×D₁₆	C₅⋊D₅
hypocenter	C₅²	#2177
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

688 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 1600

Polycyclic presentation:

2^7,5^2|b^2=df,c^2=e,d^2=fg,f^2=g,b^a=bd,c^b=cg,d^a=df,f^a=fg,h^c=h^4,i^c=i^4

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#2177, #72253159, #72253162
maximal normal subgroup of	C₅⋊D₅×#2177, #72253159, #72253162

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