Finite Group DatabasebetaFinite Group Databasebeta| order | 48000 |
| isoclinism | isoclinic class of #72227947 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C5 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C24 | #72209598 |
| Fitting, socle | C24×C52 | SL(2,𝔽5) |
| cosocle | C24×C5⋊D5 | A5 |
| LCS | G | |
| UCS | C1 | |
261 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 300 |
A minimal faithful permutation representation — 2 generators on 65 points:
(1,3,10,5)(2,9,21,15)(4,7,19,12)(6,11,13,24)(8,16,20,22)(14,23,25,17)(26,27,29,30)(28,32,34,39)(31,36,35,43)(33,40,41,50)(37,45,42,51)(46,54,52,60)(47,48)(49,57,53,61)(55,63,56,62)(58,64,59,65);(2,10,14,24,7,5,8,19,16,15)(3,4)(6,20,13,21,23,12,18,22,17,25)(9,11)(26,35,44,52,40)(27,28,47,55,45)(29,31,48,56,51)(30,34,38,46,50)(32,33,58,65,63,43,41,49,57,54)(36,37,59,61,62,39,42,53,64,60)22 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((1,3,10,5)(2,9,21,15)(4,7,19,12)(6,11,13,24)(8,16,20,22)(14,23,25,17)(26,27,29,30)(28,32,34,39)(31,36,35,43)(33,40,41,50)(37,45,42,51)(46,54,52,60)(47,48)(49,57,53,61)(55,63,56,62)(58,64,59,65), (2,10,14,24,7,5,8,19,16,15)(3,4)(6,20,13,21,23,12,18,22,17,25)(9,11)(26,35,44,52,40)(27,28,47,55,45)(29,31,48,56,51)(30,34,38,46,50)(32,33,58,65,63,43,41,49,57,54)(36,37,59,61,62,39,42,53,64,60));