Finite Group DatabasebetaFinite Group Databasebeta| order | 100 |
| isoclinism | isoclinic class of #1035 |
| automorphism | Aut(G) = AGL(1,ℤ/25) (order 500) · Inn(G) = G · Out(G) = C5 |
| holomorph | G ⋊ Aut(G) = #72228896 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | C25 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C25 | C4 |
| Frattini, socle | C5 | AGL(1,𝔽5) |
| prime radical, cosocle | D25 | C2 |
| LCS | G ▷ C25 | |
| UCS | C1 | |
9 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24);(1,2,6,16,13,3,7,4,11,22,8,19,12,23,14,20,25,17,10,18,9,21,24,15,5)Polycyclic presentation:
2^2,5^2|a^2=b,c^5=d,c^a=c^2d,d^a=d^2,c^b=c^4d^4,d^b=d^4| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 |
| order | 1 | 5 | 25 | 25 | 25 | 25 | 25 | 2 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | ||
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | ||
| χ5 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| χ6 | 4 | -1 | −ζ34−ζ38−ζ62−ζ66 | ζ6−ζ18+ζ46+ζ66−ζ74−ζ82+ζ86 | −ζ14+ζ22+ζ42+ζ62+ζ82−ζ86−ζ98 | ζ14+ζ18−ζ22+ζ34+ζ38−ζ46+ζ58+ζ74+ζ94+ζ98 | −ζ6−ζ42−ζ58−ζ94 | 0 | 0 | 0 |
| χ7 | 4 | -1 | ζ6−ζ18+ζ46+ζ66−ζ74−ζ82+ζ86 | −ζ14+ζ22+ζ42+ζ62+ζ82−ζ86−ζ98 | ζ14+ζ18−ζ22+ζ34+ζ38−ζ46+ζ58+ζ74+ζ94+ζ98 | −ζ6−ζ42−ζ58−ζ94 | −ζ34−ζ38−ζ62−ζ66 | 0 | 0 | 0 |
| χ8 | 4 | -1 | −ζ14+ζ22+ζ42+ζ62+ζ82−ζ86−ζ98 | ζ14+ζ18−ζ22+ζ34+ζ38−ζ46+ζ58+ζ74+ζ94+ζ98 | −ζ6−ζ42−ζ58−ζ94 | −ζ34−ζ38−ζ62−ζ66 | ζ6−ζ18+ζ46+ζ66−ζ74−ζ82+ζ86 | 0 | 0 | 0 |
| χ9 | 4 | -1 | ζ14+ζ18−ζ22+ζ34+ζ38−ζ46+ζ58+ζ74+ζ94+ζ98 | −ζ6−ζ42−ζ58−ζ94 | −ζ34−ζ38−ζ62−ζ66 | ζ6−ζ18+ζ46+ζ66−ζ74−ζ82+ζ86 | −ζ14+ζ22+ζ42+ζ62+ζ82−ζ86−ζ98 | 0 | 0 | 0 |
| χ10 | 4 | -1 | −ζ6−ζ42−ζ58−ζ94 | −ζ34−ζ38−ζ62−ζ66 | ζ6−ζ18+ζ46+ζ66−ζ74−ζ82+ζ86 | −ζ14+ζ22+ζ42+ζ62+ζ82−ζ86−ζ98 | ζ14+ζ18−ζ22+ζ34+ζ38−ζ46+ζ58+ζ74+ζ94+ζ98 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 100 (ζ100 = 1); i = ζ25.
| maximal subgroup of | C2×#1035, C3×#1035, C3⋊#1035, … |
| maximal normal subgroup of | C2×#1035, C3×#1035, C5×#1035, … |
| maximal quotient of | #6016, C2×#1035, C3×#1035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(100, 3);