#72221132

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×AGL(1,ℤ/8) ⋊ C₄
Sylow	C₅² ⋊ #21236

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×#1043
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1043
Fitting	C₂×C₅²×AGL(1,ℤ/8)	C₄
socle	C₂²×C₅²	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₅⋊D₅	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	#1043
hypocenter	C₅²	#21236
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

7128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|a^2=g,b^2=e,g^2=h,c^a=cgh,c^b=cf,d^a=dh,g^c=gh,i^b=i^3,i^e=i^4,j^b=j^3,j^e=j^4

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×#3151
maximal normal subgroup of	#1043×#3151
maximal quotient of	#72223274, #72223297

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);