Finite Group DatabasebetaFinite Group Databasebeta| order | 2500 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C54 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C54 | C4 |
| prime radical, cosocle | C52×C5⋊D5 | C2 |
| LCS | G ▷ C54 | |
| UCS | C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4,5)(7,8,9,10)(12,13)(14,15)(17,18)(19,20);(6,7,10,8,9)(16,17,19,20,18);(1,2,5,3,4)(11,12,14,15,13)Polycyclic presentation:
2^2,5^4|a^2=b,c^a=c^3,c^b=c^4,d^a=d^3,d^b=d^4,e^a=e^4,f^a=f^4178 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊D5×#1043, #72229031, #72229037, … |
| maximal normal subgroup of | C5⋊D5×#1043, #72229031, #72229037, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 5, 5, 5, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^3);;
gap> SetConjugate(coll, 3, 2, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);