Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| isoclinism | isoclinic class of #6053 |
| automorphism | Aut(G) = AΓL(1,𝔽25) (order 1200) · Inn(G) = G · Out(G) = D3 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C52 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52 | C8 |
| prime radical | C5⋊D5 | C4 |
| cosocle | #1043 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 | |
17 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 10 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2)(3,4,5,7,6,8,9,10);(1,3,5,9,6)Polycyclic presentation:
2^3,5^2|a^2=b,b^2=c,d^a=d^2e^4,e^a=d^2e^3,d^b=d^2,e^b=e^2,d^c=d^4,e^c=e^4| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 8 | 8 | 8 | 25 | 25 | 25 | 25 | 25 | 25 | 25 |
| order | 1 | 5 | 5 | 5 | 2 | 4 | 4 | 8 | 8 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ5 | 1 | 1 | 1 | 1 | -1 | ζ | −ζ | ζ3 | −ζ3 | ||
| χ6 | 1 | 1 | 1 | 1 | -1 | −ζ | ζ | −ζ3 | ζ3 | ||
| χ7 | 1 | 1 | 1 | 1 | -1 | ζ3 | −ζ3 | ζ | −ζ | ||
| χ8 | 1 | 1 | 1 | 1 | -1 | −ζ3 | ζ3 | −ζ | ζ | ||
| χ9 | 8 | 3 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 8 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 8 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 8 (ζ8 = 1); i = ζ2.
| maximal subgroup of | #6053⋊C2, C2×#6053, #10597167, … |
| maximal normal subgroup of | #6053⋊C2, C2×#6053, AGL(1,𝔽25), … |
| maximal quotient of | #87822, C2×#6053, C3×#6053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 40);