#72221232

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #2897

Subgroups

	subgroup	G/H
center	C₂	C₂²×#87831
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×#41	C₂×C₄
socle	C₂×C₅²	C₂²×C₂²⋊C₄
prime radical	#13107838	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#2897
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ #41

1850 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 1850 conjugacy classes of subgroups — too many to draw. Only its 245 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|a^2=g,b^2=e,c^2=e,f^2=g,b^a=bf,c^a=cfg,d^a=dg,d^b=dg,f^a=fg,f^b=fg,f^c=fg,h^a=h^4,h^b=h^3,h^c=h^3,h^e=h^4,i^a=h^3i,i^b=i^3,i^c=i^3,i^e=i^4

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223417, #72223426, #72223447, …
maximal normal subgroup of	#72223417, #72223426, #72223447, …

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