#72213689

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#120 ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72213578
commutator	C₂³×C₅²	C₂³×C₄
Frattini	C₂³	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×#120	C₂×C₄
socle	C₂×C₅²	C₄×#120
prime radical	#72213486	C₂
cosocle	C₂³×C₅⋊D₅	C₂⁴
hypercenter	#120	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×B(4,𝔽₂)
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

5044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214138
holomorph of	C₂×Dic₅

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