#72221297

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₄×#131

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂²×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂²×C₄²
Fitting	C₅²×#41	C₂×C₄
prime radical	#41×C₅⋊D₅	C₂²
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ #41

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223261, #72223366, #72223429, …
maximal normal subgroup of	#72223261, #72223366, #72223429, …
maximal quotient of	C₂×#72221297, #72242675

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