Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| isoclinism | isoclinic class of C5⋊D5 |
| automorphism | Aut(G) = C22×AGL(2,𝔽5) (order 48000) · Inn(G) = C5⋊D5 · Out(G) = C22×A5⋊C4 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C5 |
No aliases for this group.
| derived, Sylow | C52 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C5⋊D5 |
| commutator, hypocenter | C52 | C8 |
| Fitting, cosocle | C4×C52 | C2 |
| socle, prime radical | C2×C52 | C4 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,5,4,6,7,8)(10,11)(12,13)(15,16)(17,18);(14,15,17,18,16);(9,10,12,13,11)Polycyclic presentation:
2^3,5^2|a^2=b,b^2=c,d^a=d^4,e^a=e^456 conjugacy classes — view the full character table.
| derived subgroup of | #11716792, #11716793, #11716794, … |
| maximal subgroup of | #87764, #87765, D5×#168, … |
| maximal normal subgroup of | #87764, #87765, D5×#168, … |
| maximal quotient of | #87756, C2×#6029, C3×#6029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 16);