Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C5×SL(2,𝔽5) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×A5 |
| commutator, hypocenter | C5×SL(2,𝔽5) | C22 |
| Fitting | C4×C5 | C2×A5 |
| socle, cosocle | C2×C5 | C22×A5 |
| hypercenter | C4 | D5×A5 |
| LCS | G ▷ C5×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 | |
186 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,4,3,5)(6,46,30,20,27,38,19,28,17,9,13,48)(7,47,21,26,29,53,11,43,44,12,24,42)(8,52,18,41,36,39,37,50,22,16,33,40)(10,32,31,45,14,51,25,35,34,23,15,49);(2,5)(3,4)(6,46,8,43,15,51,10,53,22,38,19,28,37,47,14,49,25,42,18,48)(7,50,11,52)(9,27,16,24,35,34,23,44,39,17,20,13,41,29,32,31,45,21,40,30)(12,33,26,36)60 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3,5)(6,46,30,20,27,38,19,28,17,9,13,48)(7,47,21,26,29,53,11,43,44,12,24,42)(8,52,18,41,36,39,37,50,22,16,33,40)(10,32,31,45,14,51,25,35,34,23,15,49), (2,5)(3,4)(6,46,8,43,15,51,10,53,22,38,19,28,37,47,14,49,25,42,18,48)(7,50,11,52)(9,27,16,24,35,34,23,44,39,17,20,13,41,29,32,31,45,21,40,30)(12,33,26,36));