#72242577

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×M₁₆ ⋊ C₄
Sylow	C₅² ⋊ #1809

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₄⋊#1044
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1044
Fitting	C₂×C₅²×M₁₆	C₄
socle	C₂²×C₅²	C₂×C₄⋊C₄
prime radical	C₂³×C₅⋊D₅	D₄
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×M₁₆	#1044
hypocenter	C₅²	#1809
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

930 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 116 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,c^2=f,f^2=g,b^a=be,c^a=cf,c^b=cg,f^a=fg,h^a=h^2,h^d=h^4,i^a=h^3i^3,i^d=i^4

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242587, #72242588, #72242608, …
maximal normal subgroup of	#72242587, #72242588, #72242608, …

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