#72223086

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅²×M₁₆ ⋊ C₂
Sylow	C₅² ⋊ #3184

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₄⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂³×C₅⋊D₅
Fitting	C₄×C₅²×M₁₆	C₂
socle	C₂²×C₅²	C₂²×D₄
hypercenter	C₄×M₁₆	C₅⋊D₅
hypocenter	C₅²	#3184
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

1784 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

normal subgroups only

This group has 1784 conjugacy classes of subgroups — too many to draw. Only its 379 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|c^2=e,d^2=fg,f^2=g,b^a=be,c^a=ce,d^a=df,d^b=dg,f^a=fg,h^a=h^4,i^a=i^4

Conjugacy classes & character table

512 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#3184, #72223224, #72223245
maximal normal subgroup of	C₅⋊D₅×#3184, #72223224, #72223245

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