#72226600

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×#127 ⋊ C₄
Sylow	C₅² ⋊ #21203

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊#1044
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₂×C₅²×#127	C₄
socle	C₂²×C₅²	C₂²×C₂²⋊C₄
prime radical	C₂×#127×C₅⋊D₅	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×#127	#1044
hypocenter	C₅²	#21203
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

7492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 22 points:

(2,4)(6,12)(7,8)(10,11)(13,14,17,15,16)(18,19,20,22,21);(6,11)(10,12);(1,2)(3,4)(6,10)(7,8)(11,12);(5,6,7,10,9,11,8,12)(14,15,16,17)(19,20,21,22)

Polycyclic presentation:

2^8,5^2|a^2=e,e^2=h,g^2=h,b^a=bg,c^a=ch,d^a=dgh,d^b=df,e^b=eh,e^d=eh,g^b=gh,g^d=gh,i^a=i^2,i^e=i^4,j^a=i^3j^3,j^e=j^4

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

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