Finite Group DatabasebetaFinite Group Databasebeta| order | 100 |
| isoclinism | isoclinic class of #1043 |
| automorphism | Aut(G) = AGL(2,𝔽5) (order 12000) · Inn(G) = G · Out(G) = S5 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C52 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52 | C4 |
| prime radical, cosocle | C5⋊D5 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 3 generators on 10 points:
(2,3,4,5)(7,8,9,10);(6,7,8,10,9);(1,2,3,5,4)Polycyclic presentation:
2^2,5^2|a^2=b,c^a=c^2,d^a=d^2,c^b=c^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 | 5 | 5 | 5 | 5 | 5 | 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 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| χ7 | 4 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 |
| χ8 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 |
| χ9 | 4 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 |
| χ10 | 4 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | #6053, D5⋊AGL(1,𝔽5), C2×#1043, … |
| maximal normal subgroup of | #6053, D5⋊AGL(1,𝔽5), C2×#1043, … |
| maximal quotient of | #6033, C2×#1043, C3×#1043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(100, 11);